Question
Find the matrix A satisfying the matrix equation:$\begin{bmatrix}2&1\\3&2\end{bmatrix}\text{A}\begin{bmatrix}-3&2\\5&-3\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$
✓
Answer
We have, $\begin{bmatrix}2&1\\3&2\end{bmatrix}\text{A}\begin{bmatrix}-3&2\\5&-3\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ Or PAQ = I, where $\text{P}=\begin{bmatrix}2&1\\3&2\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}-3&2\\5&-3\end{bmatrix}$ $\therefore\ \text{P}^{-1}\text{PAQ}=\text{P}^{-1}\text{I}$ $\Rightarrow\ \text{IAQ}=\text{P}^{-1}$ $\Rightarrow\ \text{AQ}=\text{P}^{-1}$ $\Rightarrow\ \text{AQQ}^{-1}=\text{P}^{-1}\text{Q}^{-1}$ $\Rightarrow\ \text{AI}=\text{P}^{-1}\text{Q}^{-1}$ $\Rightarrow\ \text{A}=\text{P}^{-1}\text{Q}^{-1}$ Now adj. $\text{P}=\begin{bmatrix}2&-1\\-3&2\end{bmatrix}$ and $|\text{P}|=1$ $\therefore\ \text{P}^{-1}=\begin{bmatrix}2&-1\\-3&2\end{bmatrix}$ Also adj. $\text{Q}=\begin{bmatrix}-3&-2\\-5&-3\end{bmatrix}$ and $|\text{Q}|=-1$ $\therefore\ \text{Q}^{-1}=\begin{bmatrix}3&2\\5&3\end{bmatrix}$$\Rightarrow\ \text{A}=\text{P}^{-1}\text{Q}^{-1}$
$=\begin{bmatrix}2&-1\\-3&2\end{bmatrix}\begin{bmatrix}3&2\\5&3\end{bmatrix}$
$=\begin{bmatrix}6-5&4-3\\-9+10&-6+6\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}$
$=\begin{bmatrix}2&-1\\-3&2\end{bmatrix}\begin{bmatrix}3&2\\5&3\end{bmatrix}$
$=\begin{bmatrix}6-5&4-3\\-9+10&-6+6\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}$
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