Answer
$\begin{bmatrix}4&-8\\ -7&14 \end{bmatrix}$
Work Step by Step
Step 1. List the matrices needed:
$\begin{array} \\A=\\ \end{array}
\begin{bmatrix} 2&1\\ 3&2 \end{bmatrix},
\begin{array} \\B=\\ \end{array}
\begin{bmatrix} 1&-2\\ -2&4 \end{bmatrix}$
Step 2. Solve X from the equation: $A^{-1}AX=X=A^{-1}B$
Step 3. Calculate $A^{-1}$ (use the formula from section 10.5):
$\begin{array} \\A^{-1}=\\ \end{array}
\frac{1}{2\times2-1\times3}\begin{bmatrix} 2&-1\\ -3&2 \end{bmatrix}
=\begin{bmatrix} 2&-1\\ -3&2 \end{bmatrix}$
Step 4. Perform the operation:
$\begin{array} \\A^{-1}B=\\ \end{array}
\begin{bmatrix} 2&-1\\ -3&2 \end{bmatrix}
\begin{bmatrix} 1&-2\\ -2&4 \end{bmatrix}
=\begin{bmatrix}4&-8\\ -7&14 \end{bmatrix}$
Step 5. Write the solution:
$\begin{array} \\X=\\ \end{array}
\begin{bmatrix}4&-8\\ -7&14 \end{bmatrix}$
