Exam 7: Matrices and Determinants

The area of a triangle with vertices $\left( x _ { 1 } , y _ { 1 } \right),$ $\left( x _ { 2 } , y _ { 2 } \right),$ and $\left( x _ { 3 } , y _ { 3 } \right)$ can be given as the absolute value of the determinant $\frac { 1 } { 2 } \left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\x _ { 2 } & y _ { 2 } & 1 \\x _ { 3 } & y _ { 3 } & 1\end{array} \right|$ Use this formula to find the area of each triangle whose coordinates are $( - 6,2 ),$ $( 0,3 ),$ and $( - 5,6 ).$

Write the matrix in reduced row-echelon form. $\left[ \begin{array} { c c c c } 7 & - 5 & 4 & 1 \\- 7 & 3 & - 2 & 21 \\8 & 2 & 3 & - 53\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c c } 8 & 8 & 8 \\24 & 40 & 32 \\24 & 48 & 40\end{array} \right]$ .

Given matrix $A = \left[ \begin{array} { c c } 7 & 15 \\- 14 & 5\end{array} \right]$ . Find $A ^ { - 1 }$ the inverse matrix.

Use a determinant to find the area of the triangle shown below.

Find $A B$ $A = \left[ \begin{array} { l l } - 8 & 4 \\- 4 & 7\end{array} \right]$ $B = \left[ \begin{array} { c c } - 4 & - 2 \\4 & - 4\end{array} \right]$

Find $A + B$ $A = \left[ \begin{array} { c c } - 1 & 7 \\8 & - 4 \\6 & - 1\end{array} \right] \quad B = \left[ \begin{array} { c c } - 9 & 7 \\3 & 5 \\- 3 & - 7\end{array} \right]$

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y ) , \text { and } ( 15 , - 6 )$ are collinear.

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 9 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\- 4 & - 8 & 3 \\1 & 2 & 0\end{array} \right]$ (if it exists).

If possible, find 4A + 5B. $A = \left[ \begin{array} { c c c } - 7 & 1 & - 2 \\- 7 & - 6 & 1\end{array} \right] , B = \left[ \begin{array} { c c c } 9 & 2 & 7 \\- 2 & 8 & - 7\end{array} \right]$

Evaluate the expression. $\frac { 3 } { 4 } \left[ \begin{array} { l l l } - 9 & 7 & - 1\end{array} \right] + \left[ \begin{array} { l l l } - 6 & 3 & - 9\end{array} \right]$

Write the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}x + 9 y + 7 z & = 9 \\- y + 8 z & = - 5 \\x + 2 z & = - 4\end{aligned} \right.$

Given $A = \left[ \begin{array} { c c c } - 5 & 5 & 5 \\10 & - 5 & 0 \\- 15 & 0 & - 5\end{array} \right]$ , use $A ^ { - 1 }$ to decode the following cryptogram: -355 60 -35 -185 15 -25 60 -95 -40 60 -30 10 -145 70 45 -255 90 45 -5 25 45

Two competing companies offer cable television to a city with 100,000 households. Gold Cable Company has 10,000 subscribers and Galaxy Cable Company has 15,000 subscribers. The percent changes in cable subscriptions each year are shown in the matrix below $\left[ \begin{array} { l l l } 0.80 & 0.25 & 0.20 \\0.15 & 0.50 & 0.05 \\0.05 & 0.25 & 0.75\end{array} \right]$ where the columns are percent changes from Gold, from Galaxy, and from Nonsubscriber respectively and the rows are percent changes to Gold, to Galaxy, and to Nonsubscriber respectively. Find the number of nonsubscribers in two years using matrix multiplication.

Find A2. (Note: A2 = AA.) $A = \left[ \begin{array} { c c } 4 & 5 \\- 4 & - 3\end{array} \right]$

Solve the system of equations below by the Gaussian elimination method. $8 x - 6 y - 8 z = 28$ $- 7 x - 4 y + 7 z = - 43$ $- 3 x + 3 y + 3 z = - 9$

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\left\{ \begin{aligned}x - 8 y - 8 z & = - 36 \\y + 2 z & = 7 \\z & = 2\end{aligned} \right.$ $\left\{ \begin{aligned}x - 7 y + 5 z & = - 7 \\y + 2 z & = 7 \\z & = 2\end{aligned} \right.$

Find $B A$ $A = \left[ \begin{array} { c c c } - 9 & 2 & - 6 \\9 & 3 & - 2 \\0 & - 2 & 5\end{array} \right] \quad B = \left[ \begin{array} { c c c } 9 & - 8 & 9 \\5 & 0 & 7 \\- 6 & - 7 & 0\end{array} \right]$

An augmented matrix that represents a system of linear equations (in variables x, y, and z) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $\left[ \begin{array} { c c c : c } 1 & 0 & 0 & - 3 \\0 & 1 & 0 & - 6 \\0 & 0 & 1 & 1\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c } - 3 & - 1 \\- 5 & 0\end{array} \right]$ (if it exists).