Exam 7: Linear Systems and Matrices

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { c c c } 1 & - 1 & 9 \\- 3 & 4 & 4 \\6 & - 8 & - 8\end{array} \right]$ Add 3 times $R _ { 1 }$ to $R _ { 2 }$ . Add $- 6$ times $R _ { 1 }$ to $R _ { 3 }$ .

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } 3 x - 6 y = - 8 \\9 x + 8 y = 5\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}x + y + z & = - 11 \\x - 6 y - 3 z & = 13 \\6 y - 7 z & = 42\end{aligned} \right.$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $\left\{ \begin{array} { l } x + 4 y + 8 z = 5 \\7 x - 3 y - 4 z = - 64 \\- x - 4 y - 8 z = - 5 \\6 x + 5 y + 7 z = - 18\end{array} \right.$

Use the matrix capabilities of a graphing utility to find $A B$ , if possible. $A = \left[ \begin{array} { c c c } - 7 & 5 & 2 \\0 & 1 & 8 \\2 & 8 & - 2\end{array} \right] , B = \left[ \begin{array} { c c c } 4 & 3 & - 9 \\- 4 & - 1 & - 3 \\9 & 8 & 1\end{array} \right]$

Find a solution of the system of equations and verify your result using matrix multiplication. $\left\{ \begin{aligned}- 8 x + 3 y & = 20 \\3 x - 5 y & = - 9\end{aligned} \right.$

Evaluate the expression. $\left[ \begin{array} { c c } 4 & - 3 \\- 9 & 1\end{array} \right] + \left[ \begin{array} { c c } 2 & 8 \\- 2 & - 4\end{array} \right] + \left[ \begin{array} { c c } - 2 & - 1 \\- 8 & 4\end{array} \right]$

Evaluate the expression. $\left[ \begin{array} { c } 2 \\5 \\- 3\end{array} \right] \left( \left[ \begin{array} { l l } 6 & 4\end{array} \right] + \left[ \begin{array} { l l } - 8 & 1\end{array} \right] \right)$

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { c c c } 1 & - 2 & - 3 \\2 & - 3 & - 1 \\1 & - 8 & - 33\end{array} \right]$ Add $- 2$ times $R _ { 1 }$ to $R _ { 2 }$ . Add $- 1$ times $R _ { 1 }$ to $R _ { 3 }$ .

If possible, find $5 A + 4 B$ . $A = \left[ \begin{array} { r r r } - 8 & 9 & 2 \\- 2 & 1 & - 4\end{array} \right] , B = \left[ \begin{array} { c c c } 8 & 2 & 1 \\- 3 & 8 & 4\end{array} \right]$

Find the inverse of the matrix $\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right]$ (if it exists).

If possible, find $4 A + 2 B$ . $A = \left[ \begin{array} { c c c } 0 & - 4 & 4 \\1 & 0 & - 3\end{array} \right] , B = \left[ \begin{array} { c c c } - 3 & 2 & 5 \\- 5 & 8 & - 2\end{array} \right]$

Determine the order of the matrix. $\left[ \begin{array} { c c c } - 4 & - 8 & 2 \\0 & - 7 & - 4\end{array} \right]$

Evaluate the expression. $\left[ \begin{array} { c c } - 7 & - 3 \\6 & 5\end{array} \right] + \left[ \begin{array} { c c } - 6 & 7 \\- 1 & 2\end{array} \right] + \left[ \begin{array} { c c } 4 & 9 \\9 & 4\end{array} \right]$

Solve for $x$ given the following equation involving a determinant. $\left| \begin{array} { c c } x + 1 & 7 \\- 1 & x + 9\end{array} \right| = 0$

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 289 = 0 \\4 x - 25 y - 340 = 0\end{array} \right.$

Given: $A = \left[ \begin{array} { r r r } - 6 & - 10 & 7 \\- 2 & 4 & 3\end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & 7 & - 3 \\1 & 2 & - 2\end{array} \right] , c = - 4 \text { and } d = 2 \text {, }$ determine $c A + d B$ .

Solve the system of equations below using Gaussian elimination. 5x+4y+3z =-72 x-2y+2z =-48 x-y-z =24

Solve the system of linear equations. $\left\{ \begin{array} { c } x + y + z + w = 3 \\- x - y + z + w = - 5 \\x + 3 z + w = - 6 \\- 4 x - 4 y - 2 z - w = - 13\end{array} \right.$

Find $x$ and $y$ . $\left[ \begin{array} { c c c } x + 3 & 6 & 4 y \\5 & 2 x & - 1 \\8 & y - 9 & 2\end{array} \right] = \left[ \begin{array} { c c c } 9 x + 19 & 6 & - 16 \\5 & - 4 & - 1 \\8 & - 13 & 2\end{array} \right]$