Exam 2: Matrix Algebra

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } 10 & 1 \\ - 1 & 0 \end{array} \right]$ and $\left[ \begin{array} { r r } 0 & 1 \\ - 1 & 10 \end{array} \right]$

Find the 4 × 4 matrix that produces the described transformation, using homogeneous coordinates. -Translation by the vector (4, -6, -3)72)

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { r r } - 2 & 4 \\ 4 & - 4 \end{array} \right]$ and $\left[ \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 4 } \\ \frac { 1 } { 2 } & \frac { 1 } { 4 } \end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 5 & 3 \\ 3 & 2 \end{array} \right]$ and $\left[ \begin{array} { r r } 2 & - 3 \\ - 3 & 5 \end{array} \right]$

Find the 3 × 3 matrix that produces the described transformation, using homogeneous coordinates. - $( x , y ) \rightarrow ( x + 7 , y + 4 )$

Solve the equation Ax = b by using the LU factorization given for A. - $A = \left[ \begin{array} { r r r r } 1 & 2 & 4 & 3 \\- 1 & - 3 & - 1 & - 4 \\2 & 1 & 19 & 3 \\1 & 5 & - 9 & 7\end{array} \right] , \mathbf { b } = \left[ \begin{array} { l } 2 \\0 \\4 \\3\end{array} \right]$ $A = \left[ \begin{array} { r r r r } 1 & 0 & 0 & 0 \\- 1 & 1 & 0 & 0 \\2 & 3 & 1 & 0 \\1 & - 3 & - 2 & 1\end{array} \right] \left[ \begin{array} { r r r r } 1 & 2 & 4 & 3 \\0 & - 1 & 3 & - 1 \\0 & 0 & 2 & 0 \\0 & 0 & 0 & 1\end{array} \right]$

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

Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and final demand vector d. Round production levels to the nearest whole number. -Compute the matrix of the transformation that performs the shear transformation $x \rightarrow A x$ for $A = \left[ \begin{array} { l l } 1 & 0.25 \\ 0 & 1 \end{array} \right]$ and then scales all $y$ -coordinates by a factor of $0.68$ .

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$

Perform the matrix operation. -Let $A = \left[ \begin{array} { r r } - 3 & 1 \\ 0 & 2 \end{array} \right]$ . Find $5 \mathrm {~A}$ .

Perform the matrix operation. -Let $C = \left[ \begin{array} { r } 6 \\ - 2 \\ 10 \end{array} \right]$ . Find (1/2) C.

Find the inverse of the matrix A, if it exists. - $A = \left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$

Identify the indicated submatrix. - $A = \left[ \begin{array} { r r r | r } 0 & 1 & - 4 & - 5 \\ 4 & - 1 & 0 & 7 \\ \hline 2 & 5 & - 7 & 0 \end{array} \right]$ . Find $A _ { 12 }$

Perform the matrix operation. -Let $A = \left[ \begin{array} { l l } 3 & 3 \\ 2 & 4 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & 4 \\ - 1 & 6 \end{array} \right]$ . Find $4 A + B$ .

Find the 3 × 3 matrix that produces the described transformation, using homogeneous coordinates. -Reflect through the x-axis

Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and final demand vector d. Round production levels to the nearest whole number. - $\mathrm { C } = \left[ \begin{array} { l l } .4 & .3 \\.1 & .6\end{array} \right] , \mathrm { d } = \left[ \begin{array} { l } 52 \\74\end{array} \right]$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { c c } - 5 & 1 \\ - 7 & 1 \end{array} \right]$ and $\left[ \begin{array} { l } \frac { 1 } { 2 } - \frac { 1 } { 2 } \\ \frac { 7 } { 2 } - \frac { 5 } { 2 } \end{array} \right]$

Find the matrix product AB, if it is defined. - $A = \left[ \begin{array} { l l } 1 & 0 \\0 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & 2 & - 2 \\2 & - 2 & 2\end{array} \right]$

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined. - $A$ is $1 \times 4 , B$ is $4 \times 1$ .

Find the inverse of the matrix, if it exists - $A = \left[ \begin{array} { r r } 0 & - 5 \\ 6 & 3 \end{array} \right]$