Exam 11: Matrices and Determinants

Solve $\left[ \begin{array} { l l } 1 & 2 \\3 & 4\end{array} \right] \left[ \begin{array} { l } x \\y\end{array} \right] = \left[ \begin{array} { l } 1 \\2\end{array} \right]$ for $x$ and $y$

Determine which point is a solution for the system of inequalities. $\left\{ \begin{array} { l } y > 4 - x ^ { 2 } \\y \leq x\end{array} \right.$

Graph the inequality $y \leq 3 x - 1$ .

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

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

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. $\left[ \begin{array} { c c c c } 1 & 0 & 0 & 3 \\0 & 1 & 0 & 3 \\0 & 0 & 1 & 10\end{array} \right]$

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

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

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 $( A B ) \cdot ( C D )$

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. $\left\{ \begin{array} { r } 5 x + 3 y + 11 z = 37 \\3 y + z = 12 \\15 x + 9 y + 13 z = 61\end{array} \right.$

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

Determine whether the matrix is in row-echelon form, reduced row-echelon form or neither. $\left[ \begin{array} { l l l l } 1 & 1 & 0 & 3 \\0 & 1 & 0 & 3 \\2 & 0 & 1 & 1\end{array} \right]$

Given $A = \left[ \begin{array} { l l } 0 & 5\end{array} \right]$ , $B = \left[ \begin{array} { l l } - 1 & 5 \\- 2 & 3\end{array} \right]$ , $C = \left[ \begin{array} { l l l } 0 & 1 & 2 \\6 & 2 & 3\end{array} \right]$ , $D = \left[ \begin{array} { l } 1 \\5 \\0\end{array} \right]$ , find $( A B ) \cdot ( C D )$ , or explain why the operation cannot be performed.

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.

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 $3 A - 2 B$

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 $X$