Exam 8: Matrices and Determinants

Write a cryptogram for the message "TWO IF BY LAND" using the matrix $\left[ \begin{array} { c c c } 0 & - 2 & 2 \\2 & 3 & - 3 \\3 & - 2 & 0\end{array} \right]$ .Show all your work.

Find all (a) minors and (b) cofactors of the matrix $\left[ \begin{array} { c c c } - 6 & 4 & - 16 \\- 6 & - 4 & 12 \\2 & - 6 & - 12\end{array} \right]$ .Show your work.

Use a determinant and the given vertices of a triangle to find the area of the triangle.

Use the matrix capabilities of a graphing utility to find the determinant of the matrix $\left[ \begin{array} { c c c c c } - 1 & 1 & 0 & - 2 & 3 \\0 & 0 & 1 & 4 & 0 \\0 & - 2 & - 3 & - 5 & 1 \\1 & 0 & 0 & 7 & 0 \\0 & 0 & 0 & 1 & 0\end{array} \right]$ .

Use the inverse formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$ to find the inverse of the 2×2 matrix (if it exists). $\left[ \begin{array} { c c } - 12 & 3 \\5 & 2\end{array} \right]$

Evaluate the expression. $\left[ \begin{array} { c c } - 4 & 0 \\4 & - 4\end{array} \right] + \left[ \begin{array} { c c } 9 & 1 \\- 4 & - 1\end{array} \right] + \left[ \begin{array} { c c } - 11 & - 12 \\13 & 7\end{array} \right]$

Write the system of linear equations represented by the augmented matrix.Then use back-substitution to solve.(Use variables x, y, and z.) $\left[ \begin{array} { r r r r r } 1 & 9 & - 3 & \vdots&- 39 \\0 & 1 & 2 & \vdots & 0 \\0 & 0 & 1 & \vdots & 2\end{array} \right]$

Write the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}x + 7 y + 7 z & = - 6 \\3 y + 3 z & = 6 \\x + 7 z & = - 7\end{aligned} \right.$

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 3 & 0 \\0 & 4\end{array} \right]$

Find the determinant of the matrix. $\left[ \begin{array} { c c } - \frac { 1 } { 2 } & \frac { 1 } { 3 } \\- 24 & \frac { 1 } { 3 }\end{array} \right]$

Use an inverse matrix to solve (if possible) the system of linear equations. $\left\{ \begin{array} { l } 0.2 x - 0.6 y = 14.4 \\- x + 1.4 y = - 20.8\end{array} \right.$

Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix $[ A \mid B ]$ to solve for the matrix X. $\left\{ \begin{array} { r l c } x _ { 1 } - x _ { 2 } - 3 x _ { 3 } & = & 6 \\8 x _ { 1 } - 7 x _ { 2 } + 8 x _ { 3 } & = & - 104 \\6 x _ { 1 } - 6 x _ { 2 } - 8 x _ { 3 } & = & - 14\end{array} \right.$

Find the inverse of the following matrix. $\left[ \begin{array} { c c } \cos \theta & \sin \theta \\- \sin \theta & \cos \theta\end{array} \right]$

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices. $\left\{ \begin{aligned}x - y - 6 z & = - 33 \\y + 3 z & = 18 \\z & = 4\end{aligned} \right.$ $\left\{ \begin{array} { r l r } x - 7 y - 3 z & = - 57 \\y - 9 z & = - 30 \\z & = 4\end{array} \right.$

Solve for x given the following equation involving a determinant. $\left| \begin{array} { c c } x + 2 & 5 \\- 1 & x + 8\end{array} \right| = 0$

Find 6A. $A = \left[ \begin{array} { l l } 1 & - 1 \\1 & - 1\end{array} \right]$

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

​Find a value of y such that the triangle with the given vertices has an area of 4 square units. ​ ​(-1, 8), ​(0, 4), ​(-1, y) ​

Determine the order of the matrix. $\left[ \begin{array} { r r r } 6 & - 7 & 7 \\9 & 6 & 7\end{array} \right]$

Use Cramer's Rule to solve (if possible) the system of equations. $\left\{ \begin{array} { r } 6 x - 5 y = 23 \\- 13 x + 3 y = - 89\end{array} \right.$