Exam 8: Matrices and Determinants

Use the matrix capabilities of a graphing utility to evaluate the expression.Round your results to three decimal places, if necessary. $\frac { 3 } { 7 } \left[ \begin{array} { c c } - 6 & 6 \\2 & - 4\end{array} \right] + 6 \left[ \begin{array} { c c } 8 & 2 \\- 5 & - 4\end{array} \right]$

Solve the system of linear equations. $\left\{ \begin{array} { l } x + y + z = 0 \\3 x + 5 y + 4 z = 9 \\3 x + 6 y + 5 z = 6\end{array} \right.$

Find 3A. $A = \left[ \begin{array} { c c } 5 & - 2 \\2 & 3 \\- 3 & 5\end{array} \right]$

Find the inverse of the matrix. $\left[ \begin{array} { c c c } 1 & 0 & 5 \\1 & 1 & 5 \\- 5 & 1 & 1\end{array} \right]$

Find all minors of the matrix. $\left[ \begin{array} { c c } 0 & 8 \\4 & - 8\end{array} \right]$

Find A - B. $A = \left[ \begin{array} { l l l } 4 & 9 & 2 \\6 & 3 & 2 \\9 & 6 & 8\end{array} \right] , B = \left[ \begin{array} { l l l } 3 & 9 & 9 \\6 & 8 & 7 \\7 & 7 & 8\end{array} \right]$

Find the determinant of the matrix $\left[ \begin{array} { l l } - 1 & 2 \\- 6 & 4\end{array} \right]$ .

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { l } 2 x + 6 y = 10 \\2 x + 3 y = 7\end{array} \right.$

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

Find the inverse of the matrix $\left[ \begin{array} { c c } 3 & 6 \\- 9 & - 15\end{array} \right]$ .

Use the matrix capabilities of a graphing utility to evaluate the expression. $- 4 \left( \left[ \begin{array} { l l } - 6.05 & 0.86 \\- 6.63 & 2.61\end{array} \right] - \left[ \begin{array} { c c } 3.06 & 8.81 \\- 3.54 & 3.23\end{array} \right] \right)$

Select the system of linear equations represented by the following augmented matrix. $\left[ \begin{array} { c c c c } 8 & - 4 & 1 &\vdots& 14 \\16 & 0 & - 9 &\vdots& 12\end{array} \right]$

Select the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}8 x - 4 y + z & = 14 \\16 x - 9 z & = 12\end{aligned} \right.$

Use a determinant and the given vertices of a triangle to find the area of the triangle. (-5, 5), (3, 4), (4, -6)

Select the system of linear equations represented by the following augmented matrix.(Variables x, y, z, and w are used whenever applicable.) $\left[ \begin{array} { c c c c c } 1&3 & \vdots & 8 \\3&-4& \vdots &5\end{array} \right]$

Find all the cofactors of the matrix. $\left[ \begin{array} { c c c } - 4 & 0 & 2 \\4 & 4 & 3 \\3 & - 3 & 3\end{array} \right]$

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

Use matrices to find the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}5 x - 5 y & = 5 \\- 2 x - 3 y & = 18\end{aligned} \right.$

Use an inverse matrix to solve (if possible) the system of linear equations. $\left\{ \begin{array} { l l } 1.8 x - 5 y & = 4 \\28.8 x - 16 y & = 5\end{array} \right.$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $\left\{ \begin{aligned}x + 2 y + 2 z + 4 w & = 20 \\3 x + 6 y + 5 z + 12 w & = 53 \\x + 3 y - 3 z + 2 w & = - 18 \\6 x - y - z + w & = - 30\end{aligned} \right.$