Exam 11: Matrices and Determinants

Given $A = \left[ \begin{array} { c c } 1 & 5 \\- 2 & 3\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 4 & 1 & - 1 \\3 & - 5 & 2\end{array} \right]$ , find $B A$ , or explain why the operation cannot be performed.

Sketch the triangle with vertices $\left( \frac { 3 } { 2 } , 0 \right)$ , $( 2,6 )$ , and $( 1,4 )$ , and use a determinant to find its area.

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 $3 A - 2 B$

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

Write out the form of the partial fraction decomposition of $\frac { x ^ { 2 } + 5 } { x ^ { 3 } - 2 x ^ { 2 } + 4 x - 8 }$

Solve the system. $\left\{ \begin{array} { l } x + y = - 6 \\x - 2 y = 1\end{array} \right.$

Given $A = \left[ \begin{array} { c c } 3 & - 1 \\2 & 2\end{array} \right]$ and $B = \left[ \begin{array} { c c } 2 & 3 \\- 1 & 5\end{array} \right]$ , find $A B$ , or explain why the operation cannot be performed.

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?

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

Given $A = \left[ \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 2 & 4 \\3 & 8\end{array} \right]$ , solve $X - 2 A = B$ , for the matrix $X$ , or explain why there is no solution.

Write the system of equations in matrix form. $\left\{ \begin{array} { l } a - b + c = 8 \\4 a - b + c = 0 \\a + b + 2 c = 7 \\a + c + d = 0\end{array} \right.$

Find the partial fraction decomposition of $\frac { 4 } { x ^ { 3 } - 4 x ^ { 2 } + 4 x }$

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + y - 3 z = 0 \\x + y + z = 4 \\x - y = 2\end{array} \right.$

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

Find the partial fraction decomposition of $\frac { 1 } { x ^ { 3 } - 4 x }$ .

The system is linear. $\left\{ \begin{array} { r } x + \sqrt { 2 } y + z + w = e ^ { 2 } \\3 x + 2 y - z - w = 0 \\x + e ^ { 2 } y + z + 2 w = 1 \\x + y + z + w = 1\end{array} \right.$

Solve the system of equations $\left\{ \begin{array} { r } 3 x + 2 y = 8 \\x - 2 y = 0\end{array} \right.$ by converting to a matrix equation and using its inverse coefficient matrix $\left[ \begin{array} { c c } \frac { 1 } { 4 } & \frac { 1 } { 4 } \\\frac { 1 } { 3 } & - \frac { 3 } { 3 }\end{array} \right]$ .

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

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

Find the dimension of matrix A. $A = \left[ \begin{array} { c c c c c } 1 & 2 & 3 & 4 & 5 \\6 & 7 & 8 & 9 & 10\end{array} \right]$