Question
If $I$ is the unit matrix of order $2 \times 2.$ Find the matrix $M$ such that $5 M+3 I=4\left[\begin{array}{ll}2 & -5 \\ 0 & -3\end{array}\right]$
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Answer
$\begin{array}{l}5 M+3 I=4\left[\begin{array}{ll}2 & -5 \\ 0 & -3\end{array}\right] \end{array} $
$ 5 M=4\left[\begin{array}{ll}2 & -5 \\ 0 & -3\end{array}\right]-3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $
$ 5 M=\left[\begin{array}{ll}8 & -20 \\ 0 & -12\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] $
$ 5 M=\left[\begin{array}{ll}5 & -20 \\ 0 & -15\end{array}\right] $
$ M=\frac{1}{5}\left[\begin{array}{ll}5 & -20 \\ 0 & -15\end{array}\right]=\left[\begin{array}{ll}1 & -4 \\ 0 & -3\end{array}\right]$
$ 5 M=4\left[\begin{array}{ll}2 & -5 \\ 0 & -3\end{array}\right]-3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $
$ 5 M=\left[\begin{array}{ll}8 & -20 \\ 0 & -12\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] $
$ 5 M=\left[\begin{array}{ll}5 & -20 \\ 0 & -15\end{array}\right] $
$ M=\frac{1}{5}\left[\begin{array}{ll}5 & -20 \\ 0 & -15\end{array}\right]=\left[\begin{array}{ll}1 & -4 \\ 0 & -3\end{array}\right]$
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