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If $\mathrm{X}=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$, and $\mathrm{I}$ is a $2 \times 2$ identity matrix, then $\mathrm{X}^{2}-2 \mathrm{X}$ $+3$ I equals to which one of the following?
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Verified Answer
The correct answer is:
$2 X$
Given matrix is :
$\mathrm{X}=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$
$\therefore \mathrm{X}^{2}=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$
$=\left[\begin{array}{cc}1 & -2-6 \\ 0 & 9\end{array}\right]=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]$
So, the given expression is:
$\mathrm{X}^{2}-2 \mathrm{X}+3 \mathrm{I}=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]-2\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]+3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]+\left[\begin{array}{cc}-2 & +4 \\ 0 & -6\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$
$=\left[\begin{array}{cc}1-2+3 & -8+4 \\ 0 & 9-6+3\end{array}\right]=\left[\begin{array}{cc}2 & -4 \\ 0 & 6\end{array}\right]=2\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]=2 \mathrm{X}$
$\mathrm{X}=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$
$\therefore \mathrm{X}^{2}=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$
$=\left[\begin{array}{cc}1 & -2-6 \\ 0 & 9\end{array}\right]=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]$
So, the given expression is:
$\mathrm{X}^{2}-2 \mathrm{X}+3 \mathrm{I}=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]-2\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]+3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$=\left[\begin{array}{cc}1 & -8 \\ 0 & 9\end{array}\right]+\left[\begin{array}{cc}-2 & +4 \\ 0 & -6\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$
$=\left[\begin{array}{cc}1-2+3 & -8+4 \\ 0 & 9-6+3\end{array}\right]=\left[\begin{array}{cc}2 & -4 \\ 0 & 6\end{array}\right]=2\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]=2 \mathrm{X}$
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