Exam 8: Matrices and Determinants

Find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. $\left[ \begin{array} { c c c } 9 & 8 & - 2 \\2 & 2 & 0 \\- 9 & 8 & 2\end{array} \right]$

​Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix [A|B] to solve for the matrix X.Show all your work. ​ $\left\{ \begin{array} { r l r } x _ { 1 } + 8 x _ { 2 } - 3 x _ { 3 } & = 66 \\- x _ { 1 } - 9 x _ { 2 } + 8 x _ { 3 } & = - 101 \\9 x _ { 1 } + 9 x _ { 2 } + 3 x _ { 3 } & = 99\end{array} \right.$

Use a determinant to determine whether the points (-5, 1), (-8, -1) and (1, 5) are collinear.

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

If possible, find AB. $A = \left[ \begin{array} { l } - 5 \\- 1\end{array} \right] , B = \left[ \begin{array} { c c } 1 & - 2 \\- 6 & 3 \\- 6 & - 5\end{array} \right]$

Select the system of linear equations represented by the following augmented matrix. $\left[ \begin{array} { r r r r } 9 & - 6 & \vdots & 0 \\7 & 4 & \vdots & - 3\end{array} \right]$

Write the system of linear equations as a matrix equation, AX = B. $\left\{ \begin{array} { r } - x _ { 1 } + x _ { 2 } = 8 \\- 4 x _ { 1 } + x _ { 2 } = 0\end{array} \right.$

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS" by assigning a number to each letter in the alphabet such as 1 = A, 2 = B, 3 = C and so on (with 0 assigned to a blank space); Then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & - 2 & 1 \\- 1 & 1 & 1 \\1 & 2 & 0\end{array} \right]$ .

Find x and y. $\left[ \begin{array} { c c c c } 16 & 4 & 10 & 4 \\- 3 & 13 & 15 & 10 \\0 & 2 & 25 & 0\end{array} \right] = \left[ \begin{array} { c c c c } 16 & 4 & 3 x + 4 & 4 \\- 3 & 13 & 15 & 5 x \\0 & 2 & 3 y - 5 & 0\end{array} \right]$

If possible, find 3A + 2B. $A = \left[ \begin{array} { c c c } - 5 & 1 & 4 \\- 4 & 3 & - 1\end{array} \right] , B = \left[ \begin{array} { c c c } - 1 & - 3 & - 7 \\6 & - 2 & - 5\end{array} \right]$

Evaluate the determinant of the matrix below to determine which of the following makes the equation true. $\left| \begin{array} { c c } w & - 4 x \\- 9 y & z\end{array} \right| =$

Determine a positive value for y such that a triangle with vertices (-4, 2), (6, 4) and (0, y) has an area of 16 square units.

Find all cofactors of the matrix $\left[ \begin{array} { c c c } - 9 & 6 & - 24 \\- 9 & - 6 & 18 \\3 & - 9 & - 18\end{array} \right]$ .

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c } - 7 & 33 \\4 & - 19\end{array} \right]$

Find all the cofactors of the matrix. $\left[ \begin{array} { c c c } - 5 & 0 & 3 \\4 & 3 & 6 \\6 & - 6 & 6\end{array} \right]$

Given $A = \left[ \begin{array} { c c c } 0 & 2 & 4 \\- 6 & - 4 & 2 \\0 & 8 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 6 & - 4 & 0 \\2 & - 2 & 4 \\6 & 2 & 2\end{array} \right]$ , find $| B A |$ .

Find 3A. ​ $A = \left[ \begin{array} { l l } 9 & 6 \\5 & 3\end{array} \right]$

Use a determinant to find an equation of the line passing through the points. ​ (-5, 3), (1, 1) ​

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

Find all minors of the matrix $\left[ \begin{array} { c c c } 9 & - 6 & 24 \\- 9 & 6 & - 18 \\3 & - 9 & 18\end{array} \right]$ .