Exam 11: Matrices and Determinants

Given $A = \left[ \begin{array} { c c } - 9 & 8 \\- 6 & - 5 \\10 & - 16\end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \\2 & 3 \\0 & 4\end{array} \right]$ , solve $2 X - A = 3 B$ , for $X$

Find the complete solution of the system. $\left\{ \begin{array} { r } x + y + z = 2 \\2 x - 3 y + 2 z = 4 \\4 x + y - 3 z = 1\end{array} \right.$

A boat travels upstream (against the current) between two points on a river $10$ mi apart in $2.5$ hours. The return trip downstream takes only $45$ minutes. What is the boat's speed, and how fast does the current in the river flow?

Find the complete solution of the system. $\left\{ \begin{array} { l } x + z = 4 \\y + 2 z = 1 \\- y + z = 0\end{array} \right.$

Find the determinant and, if possible, the inverse of the matrix. $\left[ \begin{array} { c c c } 1 & 0 & 0 \\1 & 3 & 1 \\- 1 & - 2 & 0\end{array} \right]$

Graph the inequality $y > - x$ .

A triangle has vertices $\left( \frac { 3 } { 2 } , 0 \right)$ , $( 2,6 )$ , and $( 1,4 )$ ) Use a determinant to find its area

Find the quadratic polynomial $y = c x ^ { 2 } + b x + c$ whose graph passes through the points $( - 2 , - 16 )$ , $( - 1 , - 6 )$ , and $( 3 , - 6 )$ .

Write the system of equations for which the given matrix is the augmented matrix. $\left[ \begin{array} { c c c c } 0 & 2 & 6 & 3 \\1 & 0 & 0 & - 9 \\2 & 3 & 0 & 0\end{array} \right]$

Determine whether the system is linear or nonlinear. $\left\{ \begin{array} { l } y ^ { 2 } = 3 - 2 x \\y ^ { 2 } - 4 = 2 x\end{array} \right.$

Solve the system of equations $\left\{ \begin{array} { r } x + 3 y - 2 z = 1 \\2 x + 4 y + z = 4 \\2 x + 6 y - z = 2\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c c } \frac { 3 } { 3 } & \frac { 3 } { 2 } & - \frac { 11 } { 6 } \\- \frac { 2 } { 3 } & - \frac { 1 } { 2 } & \frac { 3 } { 6 } \\- \frac { 2 } { 3 } & 0 & \frac { 1 } { 3 }\end{array} \right]$

Find the complete solution of the system. $\left\{ \begin{array} { l } x + z = 4 \\y + 2 z = 1 \\- y + z = 0\end{array} \right.$

Find the quadratic polynomial $y = a x ^ { 2 } + b x + c$ whose graph passes through the points $( - 3,7 )$ , $( - 2,1 )$ , and $( 1,7 )$

Solve $\left[ \begin{array} { l l } 2 & x \\6 & 0\end{array} \right] \left[ \begin{array} { l l } y & 3 \\- 2 & 2\end{array} \right] = \left[ \begin{array} { l l } 1 & 3 \\- 6 & 18\end{array} \right]$ for $x$ and $y$ .

Given $A = \left[ \begin{array} { c c c } 2 & 3 & - 1 \\4 & - 2 & 0 \\- 3 & - 5 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 3 & 5 & - 3 \\5 & 2 & 2 \\- 7 & 4 & 2\end{array} \right]$ , find $3 A - 2 B$

Find inverse of the matrix. $\left[ \begin{array} { l l l } 2 & 2 & 0 \\0 & 2 & 0 \\1 & 0 & 1\end{array} \right]$

Write out the form of the partial fraction decomposition. $\frac { x ^ { 2 } - 8 } { ( x - 1 ) ^ { 2 } ( x + 3 ) }$

A boat travels upstream (against the current) between two points on a river $10$ mi apart in $2.5$ Hours. The return trip downstream takes only $45$ Minutes. What is the boat's speed, and how fast does the current in the river flow?

Solve the system. $\left\{ \begin{array} { l } x + y = 6 \\x - 2 y = 0\end{array} \right.$

Solve the system. $\left\{ \begin{array} { l } y = 3 - 2 x \\y - 4 = 2 x\end{array} \right.$