Exam 6: Matrices and Determinants

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r r r } 2 & 0 & 1 & 1 \\1 & 1 & - 1 & - 1 \\- 1 & - 2 & 1 & 0 \\0 & 0 & 1 & 1\end{array} \right] , \quad B = \left[ \begin{array} { r r r r } 2 & 0 & 0 & - 2 \\- 2 & 1 & 0 & 3 \\- 2 & 2 & 1 & 5 \\2 & - 2 & - 1 & - 3\end{array} \right]$

Encode and Decode Messages Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order. -Use the coding matrix $A = \left[ \begin{array} { l l } 2 & 1 \\ 5 & 3 \end{array} \right]$ and its inverse $A ^ { - 1 } = \left[ \begin{array} { r r } 3 & - 1 \\ - 5 & 2 \end{array} \right]$ to decode the cryptogram $\left[ \begin{array} { r r } 9 & 6 \\ 25 & 17 \end{array} \right]$ .

Write a system of linear equations in three variables, and then use matrices to solve the system. -There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer tracks sales by age group for marketing purposes. The percentage of 36- to 59-year-old buyers and the percentage of Buyers 60 and older combined exceeds the percentage of buyers 35 and younger by 38%. If the percentage Of buyers in the oldest group is doubled, it is 24% less than the percentage of users in the middle group. Find the percentage of buyers in each of the three age groups.

Perform the matrix row operation (or operations)and write the new matrix. - open bracket 1 1 -1 1 2 0 -1 1 -3 0 5 0 -4 -5 5 -2 4 0 2 -1 close bracket -4+ 2+

Solve the system using the inverse that is given for the coefficient matrix. - $x + 2 y + 3 z = - 3$ $x + y + z = 12$ $2 x + 2 y + z = 6$$\quad$ The inverse of $\left[ \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right]$ is $\left[ \begin{array} { r r r } - 1 & 4 & - 1 \\ 1 & - 5 & 2 \\ 0 & 2 & - 1 \end{array} \right]$

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r r } 1 & 0 & 0 \\- 1 & 1 & 0 \\1 & 1 & 1\end{array} \right]$

Solve the matrix equation for X. -Let $A = \left[ \begin{array} { r r } 1 & 4 \\ 0 & - 2 \\ 6 & - 6 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 8 & - 6 \\ - 1 & 4 \\ 0 & 5 \end{array} \right] ; \quad B - X = 3 A$

Apply Gaussian Elimination to Systems with More Variables than Equations Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. - x+y+z=7 x-y+2z=7

Solve the matrix equation for X. -Let $\mathrm { A } = \left[ \begin{array} { r r r } 5 & 1 & - 4 \\ 5 & 0 & 0 \\ 1 & - 5 & 4 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r r } - 1 & - 5 & - 4 \\ 0 & 1 & 1 \\ 5 & 0 & 5 \end{array} \right] ; \quad 4 \mathrm {~B} - 4 \mathrm {~A} = \mathrm { X }$

Use Matrices and Gaussian Elimination to Solve Systems Solve the system of equations using matrices. Use Gaussian elimination with back-substitution. - x+y+z =-5 x-y+4z =-13 4x+y+z =-2

Use Cramer's rule to solve the system. - 11+3=1 5+2=4

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r } 0 & - 3 \\- 5 & - 5\end{array} \right]$

Use Cramer's rule to solve the system. - $6 x + 5 y = 2$ $5 x + y = - 11$

Apply Gaussian Elimination to Systems Without Unique Solutions Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. - \[\begin{array} { l } x - y + z - w = 10 \\ \quad - 2 x + 3 y + 5 w = - 28 \\ x + 2 y + 8 z + 3 w = - 10 \\ x - 4 y - 6 z - 5 w = 30 \\

Use Matrices and Gaussian Elimination to Solve Systems Solve the system of equations using matrices. Use Gaussian elimination with back-substitution. - x-y+4z =20 5x+z =4 x+3y+z =-8

Evaluate the determinant. - $\left| \begin{array} { c c } \frac { 1 } { 7 } & - \frac { 1 } { 10 } \\ 10 & 7 \end{array} \right|$

Find the product AB, if possible. - $A=\left[\begin{array}{rrr}-2 & 7 & -7 \\ 5 & 9 & -5\end{array}\right], B=\left[\begin{array}{r}7 \\ -2 \\ -1\end{array}\right]$

Use Cramer's rule to solve the system. - 2x=19-3y 5y=-11+2x

Apply Gaussian Elimination to Systems Without Unique Solutions Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. - x+y+z+w= 7 3x-2z+5w= 11 -4x+3y+w= 4 -x-y-z-w= 6

Evaluate the determinant. - $\left| \begin{array} { l l l l } 0 & 7 & 7 & 3 \\0 & 2 & 5 & 6 \\5 & 7 & 9 & 4 \\6 & 2 & 2 & 4\end{array} \right|$