Exam 11: Matrices and Determinants

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 $y$

Find the complete solution of the system. $\left\{ \begin{array} { r } 2 x + 3 y = 11 \\3 x + 3 y = 9 \\x + y + 3 z = 1\end{array} \right.$

Find the partial fraction decomposition of $\frac { x + 15 } { x ^ { 2 } + 3 x }$ .

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

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 $\left[ \begin{array} { c c } \frac { 1 } { 4 } & \frac { 1 } { 4 } \\\frac { 1 } { 3 } & - \frac { 3 } { 3 }\end{array} \right]$ .

Which point is a solution to the inequality $y \leq 3 x - 1$ ?

A man invests a total of $\$ 21,000$ in two accounts, one paying $5.5 \%$ simple interest per year and the other paying $8.5 \%$ simple interest per year. His annual interest from both accounts is $\$ 1,280$ . How much did he invest at each rate?

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

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

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

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , $D = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$ , find $( A B ) \cdot ( C D )$

Find the determinant of the matrix $\left[ \begin{array} { r r } \frac { 1 } { 2 } & 1 \\- \frac { 1 } { 2 } & \frac { 3 } { 2 }\end{array} \right]$ , if it exists

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 $X$ , or explain why there is no solution.

Use Cramer's Rule to solve the system. $\left\{ \begin{array} { r } x + 2 y - 4 z = 0 \\2 x + 5 y = 4 \\y - z = 3\end{array} \right.$

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$ , $D = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , find $( A B ) \cdot ( C D )$

Find the dimension of matrix A. $A = \left[ \begin{array} { l } 1 \\2 \\3 \\4\end{array} \right]$

Write the system of equations in matrix form. $\left\{ \begin{array} { l } a - 3 b + c = 7 \\4 a - 1 b + c = 1 \\a + b + 2 c = 7 \\a + c + a = 0\end{array} \right.$

Graph the solution of the system of inequalities. $\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\x ^ { 2 } + y ^ { 2 } \leq 9\end{array} \right.$

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

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