Question 1

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
-$$\left[ \begin{array} { c r } 
12 & - 6 \
- 6 & 7
\end{array} \right]$$

Accepted Answer

A)   P=[32−23],D=[16003]P = \left[ \begin{array} { r r } 3 & 2 \- 2 & 3\end{array} ight] , D = \left[ \begin{array} { r r } 16 & 0 \0 & 3\end{array} ight]P=[3−2​23​],D=[160​03​] 
B)   P=[3/13−2/13−2/133/13],D=[30016]P = \left[ \begin{array} { r r } 3 / \sqrt { 13 } & - 2 / \sqrt { 13 } \- 2 / \sqrt { 13 } & 3 / \sqrt { 13 }\end{array} ight] , D = \left[ \begin{array} { r r } 3 & 0 \0 & 16\end{array} ight]P=[3/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​−2/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​−2/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​3/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​],D=[30​016​] 
C)   P=[3/132/13−2/133/13],D=[16003]P = \left[ \begin{array} { r r } 3 / \sqrt { 13 } & 2 / \sqrt { 13 } \- 2 / \sqrt { 13 } & 3 / \sqrt { 13 }\end{array} ight] , D = \left[ \begin{array} { r r } 16 & 0 \0 & 3\end{array} ight]P=[3/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​−2/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​2/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​3/13c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​],D=[160​03​] 
D)   P=[3−2−23],D=[16003]P = \left[ \begin{array} { r r } 3 & - 2 \- 2 & 3\end{array} ight] , D = \left[ \begin{array} { r r } 16 & 0 \0 & 3\end{array} ight]P=[3−2​−23​],D=[160​03​] C

Question 2

$$\text { Find the maximum value of } Q ( x ) \text { subject to the constraint } _x T _ { x = 1 } \text {. }$$
-$$Q ( x ) = 2 x _ { 1 } ^ { 2 } + 9 x _ { 2 } ^ { 2 } + 5 \times \frac { 2 } { 3 }$$

Accepted Answer

A)   2
B)   18
C)   5
D)   9  D

Question 3

Find the maximum value of $Q ( x )$ subject to the constraints $_x  T _ { x } = 1$ and $_x  { T } \mathbf { _u } = 0$, where $u$ is a unit eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form.
-$$Q ( x ) = 14 x _ { 1 } ^ { 2 } + 14 x _ { 2 } ^ { 2 } + 18 x _ { 3 } ^ { 2 } + 26 x _ { 1 } x _ { 2 } + 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$$

Accepted Answer

A)   36
B)   9
C)   1
D)   16  B

Question 4

Find a singular value decomposition of the matrix A.
-$$A = \left[ \begin{array} { r r } 
- 9 & 0 \
0 & 6
\end{array} \right]$$

Accepted Answer

A)   A=[1001][−9006][1001]A = \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} ight] \left[ \begin{array} { r r } - 9 & 0 \ 0 & 6 \end{array} ight] \left[ \begin{array} { l l } 1 & 0 \ 0 & 1 \end{array} ight]A=[10​01​][−90​06​][10​01​] 
B)   A=[−9006][9006][1001]A = \left[ \begin{array} { r r } - 9 & 0 \0 & 6\end{array} ight] \left[ \begin{array} { l l } 9 & 0 \0 & 6\end{array} ight] \left[ \begin{array} { l l } 1 & 0 \0 & 1\end{array} ight]A=[−90​06​][90​06​][10​01​] 
C)   A=[−1001][9006][1001]A = \left[ \begin{array} { r r } - 1 & 0 \0 & 1\end{array} ight] \left[ \begin{array} { l l } 9 & 0 \0 & 6\end{array} ight] \left[ \begin{array} { l l } 1 & 0 \0 & 1\end{array} ight]A=[−10​01​][90​06​][10​01​] 
D)   A=[−9006][9006][−1001]A = \left[ \begin{array} { r r } - 9 & 0 \0 & 6\end{array} ight] \left[ \begin{array} { l l } 9 & 0 \0 & 6\end{array} ight] \left[ \begin{array} { r r } - 1 & 0 \0 & 1\end{array} ight]A=[−90​06​][90​06​][−10​01​]

Question 5

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product
term. Give P and the new quadratic form.
-$$Q ( x ) = 13 x _ { 1 } ^ { 2 } + 12 x _ { 2 } ^ { 2 } + 5 x _ { 3 } ^ { 2 } + 20 x _ { 1 } x _ { 2 } + 8 x _ { 1 } x _ { 3 } + 12 x _ { 2 } x _ { 3 }$$

Accepted Answer

A)    P=[−2/3−2/31/3−2/31/3−2/31/3−2/32/3];5y12+2y22+y32\mathrm { P } = \left[ \begin{array} { r r r } - 2 / 3 & - 2 / 3 & 1 / 3 \ - 2 / 3 & 1 / 3 & - 2 / 3 \ 1 / 3 & - 2 / 3 & 2 / 3 \end{array} ight] ; 5 \mathrm { y } _ { 1 } ^ { 2 } + 2 \mathrm { y } _ { 2 } ^ { 2 } + \mathrm { y } _ { 3 } ^ { 2 }P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​−2/3−2/31/3​−2/31/3−2/3​1/3−2/32/3​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​;5y12​+2y22​+y32​ 
B)   P=[2/3−2/31/32/31/3−2/31/32/32/3];25y12+4y22+y32\mathrm { P } = \left[ \begin{array} { r r r } 2 / 3 & - 2 / 3 & 1 / 3 \2 / 3 & 1 / 3 & - 2 / 3 \1 / 3 & 2 / 3 & 2 / 3\end{array} ight] ; 25 \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 } + \mathrm { y } _ { 3 } ^ { 2 }P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​2/32/31/3​−2/31/32/3​1/3−2/32/3​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​;25y12​+4y22​+y32​ 
C)   P=[2−2121−2122];y12+4y22+25y32\mathrm { P } = \left[ \begin{array} { r r r } 2 & - 2 & 1 \2 & 1 & - 2 \1 & 2 & 2\end{array} ight] ; \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 } + 25 \mathrm { y } _ { 3 } ^ { 2 }P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​221​−212​1−22​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​;y12​+4y22​+25y32​ 
D)   P=[2−2121−2122];25y12+4y22+y32P = \left[ \begin{array} { r r r } 2 & - 2 & 1 \2 & 1 & - 2 \1 & 2 & 2\end{array} ight] ; 25 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 } + y _ { 3 } ^ { 2 }P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​221​−212​1−22​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​;25y12​+4y22​+y32​

Question 6

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix.
-$$\mathrm { S } = \left[ \begin{array} { r c c } 
30 & - 12 & 12 \
- 12 & 6 & 0 \
12 & 0 & 36
\end{array} \right]$$

Accepted Answer

A)    79.1%79.1 \%79.1% 
B)    41.7%41.7 \%41.7% 
C)    34.4%34.4 \%34.4% 
D)    65.0%65.0 \%65.0%

Question 7

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
-$$\left[ \begin{array} { r r r } 
13 & 10 & 4 \
10 & 12 & 6 \
4 & 6 & 5
\end{array} \right]$$

Accepted Answer

A)   P=[2−2121−2122],D=[2500040001]\mathrm { P } = \left[ \begin{array} { r r r } 2 & - 2 & 1 \ 2 & 1 & - 2 \ 1 & 2 & 2 \end{array} ight] , \mathrm { D } = \left[ \begin{array} { r r r } 25 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array} ight]P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​221​−212​1−22​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​,D=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​2500​040​001​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
B)   P=[2/3−2/31/32/31/3−2/31/32/32/3],D=[1000400025]P = \left[ \begin{array} { r r r } 2 / 3 & - 2 / 3 & 1 / 3 \2 / 3 & 1 / 3 & - 2 / 3 \1 / 3 & 2 / 3 & 2 / 3\end{array} ight] , D = \left[ \begin{array} { c c c } 1 & 0 & 0 \0 & 4 & 0 \0 & 0 & 25\end{array} ight]P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​2/32/31/3​−2/31/32/3​1/3−2/32/3​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​,D=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​100​040​0025​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
C)   P=[2/3−2/31/32/31/3−2/31/32/32/3],D=[2500040001]P = \left[ \begin{array} { r r r } 2 / 3 & - 2 / 3 & 1 / 3 \2 / 3 & 1 / 3 & - 2 / 3 \1 / 3 & 2 / 3 & 2 / 3\end{array} ight] , D = \left[ \begin{array} { r r r } 25 & 0 & 0 \0 & 4 & 0 \0 & 0 & 1\end{array} ight]P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​2/32/31/3​−2/31/32/3​1/3−2/32/3​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​,D=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​2500​040​001​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
D)   P=[2−2121−2122],D=[1000400025]P = \left[ \begin{array} { r r r } 2 & - 2 & 1 \2 & 1 & - 2 \1 & 2 & 2\end{array} ight] , D = \left[ \begin{array} { c c c } 1 & 0 & 0 \0 & 4 & 0 \0 & 0 & 25\end{array} ight]P=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​221​−212​1−22​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​,D=H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​100​040​0025​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​

Question 8

Find the matrix of the quadratic form.
-$$6 x _ { 1 } x _ { 2 } - 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$$

Accepted Answer

A)   [06−186018−18180]\left[ \begin{array} { r r r } 0 & 6 & - 18 \ 6 & 0 & 18 \ - 18 & 18 & 0 \end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​06−18​6018​−18180​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
B)   [3000−90009]\left[\begin{array}{rrr}3 & 0 & 0 \0 & -9 & 0 \0 & 0 & 9\end{array}ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​300​0−90​009​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
C)   [03−9309−990]\left[\begin{array}{rrr}0 & 3 & -9 \3 & 0 & 9 \-9 & 9 & 0\end{array}ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​03−9​309​−990​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
D)   [13−9319−991]\left[ \begin{array} { r r r } 1 & 3 & - 9 \3 & 1 & 9 \- 9 & 9 & 1\end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​13−9​319​−991​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​

Question 9

Find a singular value decomposition of the matrix A.
-$$A = \left[ \begin{array} { r r r } 
4 & 0 & - 4 \
4 & 4 & 4
\end{array} \right]$$

Accepted Answer

A)   A=[0−110][48000320][1/31/31/3−1/201/21/6−2/61/6]A = \left[ \begin{array} { r r } 0 & - 1 \1 & 0\end{array} ight] \left[ \begin{array} { r r r } 48 & 0 & 0 \0 & 32 & 0\end{array} ight] \left[ \begin{array} { c c c } 1 / \sqrt { 3 } & 1 / \sqrt { 3 } & 1 / \sqrt { 3 } \- 1 / \sqrt { 2 } & 0 & 1 / \sqrt { 2 } \1 / \sqrt { 6 } & - 2 / \sqrt { 6 } & 1 / \sqrt { 6 }\end{array} ight]A=[01​−10​][480​032​00​]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​0−2/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
B)   A=[0−110][43000420][1/31/31/3−1/201/21/6−2/61/6]A = \left[ \begin{array} { r r } 0 & - 1 \1 & 0\end{array} ight] \left[ \begin{array} { c c c } 4 \sqrt { 3 } & 0 & 0 \0 & 4 \sqrt { 2 } & 0\end{array} ight] \left[ \begin{array} { c c c } 1 / \sqrt { 3 } & 1 / \sqrt { 3 } & 1 / \sqrt { 3 } \- 1 / \sqrt { 2 } & 0 & 1 / \sqrt { 2 } \1 / \sqrt { 6 } & - 2 / \sqrt { 6 } & 1 / \sqrt { 6 }\end{array} ight]A=[01​−10​][43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​0​042c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​00​]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​0−2/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
C)   A=[0−110][43000420][1/3−1/21/61/30−2/61/31/21/6]A = \left[ \begin{array} { r r } 0 & - 1 \1 & 0\end{array} ight] \left[ \begin{array} { c c c } 4 \sqrt { 3 } & 0 & 0 \0 & 4 \sqrt { 2 } & 0\end{array} ight] \left[ \begin{array} { c c c } 1 / \sqrt { 3 } & - 1 / \sqrt { 2 } & 1 / \sqrt { 6 } \1 / \sqrt { 3 } & 0 & - 2 / \sqrt { 6 } \1 / \sqrt { 3 } & 1 / \sqrt { 2 } & 1 / \sqrt { 6 }\end{array} ight]A=[01​−10​][43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​0​042c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​00​]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​01/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​−2/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/6c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
D)   A=[0−110][430042][111−1011−21]A = \left[ \begin{array} { c c } 0 & - 1 \1 & 0\end{array} ight] \left[ \begin{array} { c c } 4 \sqrt { 3 } & 0 \0 & 4 \sqrt { 2 }\end{array} ight] \left[ \begin{array} { r r r } 1 & 1 & 1 \- 1 & 0 & 1 \1 & - 2 & 1\end{array} ight]A=[01​−10​][43c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​0​042c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​1−11​10−2​111​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​

Question 10

$$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$$
-$$A = \left[ \begin{array} { r r } 
2 & 0 \
0 & - 9
\end{array} \right] , x = \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 }
\end{array} \right]$$

Accepted Answer

A)    2x12−9x222 x { } _ { 1 } ^ { 2 } - 9 x \frac { 2 } { 2 }2x12​−9x22​ 
B)    2x1−9x22 x _ { 1 } - 9 x _ { 2 }2x1​−9x2​ 
C)    4x1−18x24 x1 - 18 x 24x1−18x2 
D)    −9x21+2x22- 9 x \frac { 2 } { 1 } + 2 x \frac { 2 } { 2 }−9x12​+2x22​

Question 11

Find the matrix of the quadratic form.
-$$4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$$

Accepted Answer

A)   [4−21/2−2501/203]\left[ \begin{array} { c r c } 4 & - 2 & 1 / 2 \ - 2 & 5 & 0 \ 1 / 2 & 0 & 3 \end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​4−21/2​−250​1/203​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
B)   [4−20−251/201/23]\left[ \begin{array} { r r r } 4 & - 2 & 0 \ - 2 & 5 & 1 / 2 \ 0 & 1 / 2 & 3 \end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​4−20​−251/2​01/23​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
C)   [4−40−451013]\left[ \begin{array} { r r r } 4 & - 4 & 0 \ - 4 & 5 & 1 \ 0 & 1 & 3 \end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​4−40​−451​013​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​ 
D)   [400050003]\left[ \begin{array} { l l l } 4 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 3 \end{array} ight]H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"/>​400​050​003​M347 1759 V0 H263 V1759 v0 v1759 h84z"/>​

Question 12

Find the maximum value of $Q ( x )$ subject to the constraints $_x  T _ { x } = 1$ and $_x  { T } \mathbf { _u } = 0$, where $u$ is a unit eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form.
-$$Q ( x ) = 3 x { } _ { 1 } ^ { 2 } + 8 x { } _ { 2 } ^ { 2 } + 4x \frac { 2 } { 3 }$$

Accepted Answer

A)   0
B)   3
C)   4
D)   8

Question 13

Find a singular value decomposition of the matrix A.
-$$A = \left[ \begin{array} { r r } 
4 & - 1 \
- 1 & 4
\end{array} \right]$$

Accepted Answer

A)    A=[−1/21/21/21/2][9009][−1/21/21/21/2]A = \left[ \begin{array} { r r } - 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \ 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \end{array} ight] \left[ \begin{array} { l l } 9 & 0 \ 0 & 9 \end{array} ight] \left[ \begin{array} { r r } - 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \ 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \end{array} ight]A=[−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​][90​09​][−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​] 
B)   A=[−1111][4−1−14][−1111]A = \left[ \begin{array} { r r } - 1 & 1 \1 & 1\end{array} ight] \left[ \begin{array} { r r } 4 & - 1 \- 1 & 4\end{array} ight] \left[ \begin{array} { r r } - 1 & 1 \1 & 1\end{array} ight]A=[−11​11​][4−1​−14​][−11​11​] 
C)   A=[−1111][5003][−1111]A = \left[ \begin{array} { r r } - 1 & 1 \1 & 1\end{array} ight] \left[ \begin{array} { l l } 5 & 0 \0 & 3\end{array} ight] \left[ \begin{array} { r r } - 1 & 1 \1 & 1\end{array} ight]A=[−11​11​][50​03​][−11​11​] 
D)   A=[−1/21/21/21/2][5003][−1/21/21/21/2]\mathrm { A } = \left[ \begin{array} { r r } - 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \1 / \sqrt { 2 } & 1 / \sqrt { 2 }\end{array} ight] \left[ \begin{array} { l l } 5 & 0 \0 & 3\end{array} ight] \left[ \begin{array} { r r } - 1 / \sqrt { 2 } & 1 / \sqrt { 2 } \1 / \sqrt { 2 } & 1 / \sqrt { 2 }\end{array} ight]A=[−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​][50​03​][−1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​]

Question 14

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix.
-$$\mathrm { S } = \left[ \begin{array} { r c } 
60 & - 55 \
- 55 & 280
\end{array} \right]$$

Accepted Answer

A)    89.7%89.7 \%89.7% 
B)    82.4%82.4 \%82.4% 
C)    86.2%86.2 \%86.2% 
D)    13.8%13.8 \%13.8%

Question 15

Determine whether the matrix is symmetric.
-$$\left[ \begin{array} { l l } 
1 & 6 \
5 & 0
\end{array} \right]$$

Accepted Answer

A)   Yes
B)   No

Question 16

$$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$$
-$$A = \left[ \begin{array} { r r r } 
4 & 8 & 0 \
8 & 1 & 2 \
0 & 2 & - 2
\end{array} \right] , x = \left[ \begin{array} { l } 
x _ { 1 } \
x _ { 2 } \
x _ { 3 }
\end{array} \right]$$

Accepted Answer

A)    4x12+x22−2x32+8x1+10x2+2x34 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 8 x _ { 1 } + 10 x _ { 2 } + 2 x _ { 3 }4x12​+x22​−2x32​+8x1​+10x2​+2x3​ 
B)    4×21+x22−2×234 	imes \frac { 2 } { 1 } + x _ { 2 } ^ { 2 } - 2 	imes \frac { 2 } { 3 }4×12​+x22​−2×32​ 
C)    4x12−x22+2x32−16x1x2−4x2x34 x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 } + 2 x _ { 3 } ^ { 2 } - 16 x _ { 1 } x _ { 2 } - 4 x _ { 2 } x _ { 3 }4x12​−x22​+2x32​−16x1​x2​−4x2​x3​ 
D)    4x12+x22−2x32+16x1x2+4x2x34 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - 2 x _ { 3 } ^ { 2 } + 16 x _ { 1 } x _ { 2 } + 4 x _ { 2 } x _ { 3 }4x12​+x22​−2x32​+16x1​x2​+4x2​x3​

Question 17

Determine whether the matrix is symmetric.
-$$\left[ \begin{array} { r r r } 
3 & 9 & 4 \
9 & - 5 & - 2 \
4 & - 2 & 0
\end{array} \right]$$

Accepted Answer

A)   Yes
B)    No\mathrm { No }No

Question 18

Find the singular values of the matrix.
-$$\left[ \begin{array} { r r } 
- 8 & 0 \
0 & 6
\end{array} \right]$$

Accepted Answer

A)   8,6
B)   64,36
C)    −8,6- 8,6−8,6 
D)    −64,36- 64,36−64,36

Question 19

Find the singular values of the matrix.
-$\left[ \begin{array} { r r r } 1 & 5 & 5 \ 5 & 1 & - 5 \end{array} \right]$

Accepted Answer

A)   36,66
B)    6,666 , \sqrt { 66 }6,66c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
C)    0,6,660,6 , \sqrt { 66 }0,6,66c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
D)    0,36,660,36,660,36,66

Question 20

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product
term. Give P and the new quadratic form.
-$$Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$$

Accepted Answer

A)   P=[1−221];9y1+4y2\mathrm { P } = \left[ \begin{array} { r r } 1 & - 2 \2 & 1\end{array} ight] ; 9 \mathrm { y } _ { 1 } + 4 \mathrm { y } _ { 2 }P=[12​−21​];9y1​+4y2​ 
B)   P=[1/5−2/52/51/5];9y12+4y22\mathrm { P } = \left[ \begin{array} { c c } 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \2 / \sqrt { 5 } & 1 / \sqrt { 5 }\end{array} ight] ; 9 \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 }P=[1/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​2/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​−2/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​];9y12​+4y22​ 
C)   P=[1−221];9y12+4y22P = \left[ \begin{array} { r r } 1 & - 2 \2 & 1\end{array} ight] ; 9 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 }P=[12​−21​];9y12​+4y22​ 
D)   P=[−1/5−2/52/51/5];−9y12−4y22P = \left[ \begin{array} { r r } - 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \2 / \sqrt { 5 } & 1 / \sqrt { 5 }\end{array} ight] ; - 9 y _ { 1 } ^ { 2 } - 4 y _ { 2 } ^ { 2 }P=[−1/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​2/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​−2/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1/5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​];−9y12​−4y22​

Exam 7: Symmetric Matrices and Quadratic Forms

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { c r } 12 & - 6 \\- 6 & 7\end{array} \right]$

$\text { Find the maximum value of } Q ( x ) \text { subject to the constraint } _x T _ { x = 1 } \text {. }$ - $Q ( x ) = 2 x _ { 1 } ^ { 2 } + 9 x _ { 2 } ^ { 2 } + 5 \times \frac { 2 } { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r } - 9 & 0 \\0 & 6\end{array} \right]$

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - $\mathrm { S } = \left[ \begin{array} { r c c } 30 & - 12 & 12 \\- 12 & 6 & 0 \\12 & 0 & 36\end{array} \right]$

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { r r r } 13 & 10 & 4 \\10 & 12 & 6 \\4 & 6 & 5\end{array} \right]$

Find the matrix of the quadratic form. - $6 x _ { 1 } x _ { 2 } - 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r r } 4 & 0 & - 4 \\4 & 4 & 4\end{array} \right]$

$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$ - $A = \left[ \begin{array} { r r } 2 & 0 \\0 & - 9\end{array} \right] , x = \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 }\end{array} \right]$

Find the matrix of the quadratic form. - $4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r } 4 & - 1 \\- 1 & 4\end{array} \right]$

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - $\mathrm { S } = \left[ \begin{array} { r c } 60 & - 55 \\- 55 & 280\end{array} \right]$

Determine whether the matrix is symmetric. - $\left[ \begin{array} { l l } 1 & 6 \\5 & 0\end{array} \right]$

$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$ - $A = \left[ \begin{array} { r r r } 4 & 8 & 0 \\8 & 1 & 2 \\0 & 2 & - 2\end{array} \right] , x = \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 }\end{array} \right]$

Determine whether the matrix is symmetric. - $\left[ \begin{array} { r r r } 3 & 9 & 4 \\9 & - 5 & - 2 \\4 & - 2 & 0\end{array} \right]$

Find the singular values of the matrix. - $\left[ \begin{array} { r r } - 8 & 0 \\0 & 6\end{array} \right]$

Find the singular values of the matrix. - $\left[ \begin{array} { r r r } 1 & 5 & 5 \\ 5 & 1 & - 5 \end{array} \right]$

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product term. Give P and the new quadratic form. - $Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$

Exam 7: Symmetric Matrices and Quadratic Forms

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - [12−6−67]\left[ \begin{array} { c r } 12 & - 6 \\- 6 & 7\end{array} \right][12−6​−67​]

Find a singular value decomposition of the matrix A. - A=[−9006]A = \left[ \begin{array} { r r } - 9 & 0 \\0 & 6\end{array} \right]A=[−90​06​]

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - S=[30−1212−126012036]\mathrm { S } = \left[ \begin{array} { r c c } 30 & - 12 & 12 \\- 12 & 6 & 0 \\12 & 0 & 36\end{array} \right]S=​30−1212​−1260​12036​​

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - [1310410126465]\left[ \begin{array} { r r r } 13 & 10 & 4 \\10 & 12 & 6 \\4 & 6 & 5\end{array} \right]​13104​10126​465​​

Find the matrix of the quadratic form. - 6x1x2−18x1x3+18x2x36 x _ { 1 } x _ { 2 } - 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }6x1​x2​−18x1​x3​+18x2​x3​

Find a singular value decomposition of the matrix A. - A=[40−4444]A = \left[ \begin{array} { r r r } 4 & 0 & - 4 \\4 & 4 & 4\end{array} \right]A=[44​04​−44​]

Find the matrix of the quadratic form. - 4x12+5x22+3x32−4x1x2+x2x34 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }4x12​+5x22​+3x32​−4x1​x2​+x2​x3​

Find a singular value decomposition of the matrix A. - A=[4−1−14]A = \left[ \begin{array} { r r } 4 & - 1 \\- 1 & 4\end{array} \right]A=[4−1​−14​]

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - S=[60−55−55280]\mathrm { S } = \left[ \begin{array} { r c } 60 & - 55 \\- 55 & 280\end{array} \right]S=[60−55​−55280​]

Determine whether the matrix is symmetric. - [1650]\left[ \begin{array} { l l } 1 & 6 \\5 & 0\end{array} \right][15​60​]

Determine whether the matrix is symmetric. - [3949−5−24−20]\left[ \begin{array} { r r r } 3 & 9 & 4 \\9 & - 5 & - 2 \\4 & - 2 & 0\end{array} \right]​394​9−5−2​4−20​​

Find the singular values of the matrix. - [−8006]\left[ \begin{array} { r r } - 8 & 0 \\0 & 6\end{array} \right][−80​06​]

Find the singular values of the matrix. - [15551−5]\left[ \begin{array} { r r r } 1 & 5 & 5 \\ 5 & 1 & - 5 \end{array} \right][15​51​5−5​]

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product term. Give P and the new quadratic form. - Q(x) =5x12+8x22+4x1x2Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }Q(x) =5x12​+8x22​+4x1​x2​