If A= 2&3 -1&0 , show that A^2 - 2A + 3I_2 = 0. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}2&3\\-1&0\end{bmatrix},$ show that $A^2 - 2A + 3I_2 = 0.$

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Answer

Given: $\text{A}=\begin{bmatrix}2&3\\-1&0\end{bmatrix}$
Now,
$\text{A}^2=\text{AA}$
$\Rightarrow\text{A}^2=\begin{bmatrix}2&3\\-1&0\end{bmatrix}\begin{bmatrix}2&3\\-1&0\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}4-3&6+0\\-2+0&-3+0\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}1&6\\-2&-3\end{bmatrix}$
$ \text{A}^2-2\text{A}+3\text{I}_2$
$\Rightarrow\text{A}^2-2\text{A}+3\text{I}_2=\begin{bmatrix}1&6\\-2&-3\end{bmatrix}-2\begin{bmatrix}2&3\\-1&0\end{bmatrix}+3\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$ \Rightarrow\text{A}^2-2\text{A}+3\text{I}_2=\begin{bmatrix}1&6\\-2&-3\end{bmatrix}-\begin{bmatrix}4&6\\-2&0\end{bmatrix}+\begin{bmatrix}3&0\\0&3\end{bmatrix}$
$ \Rightarrow\text{A}^2-2\text{A}+3\text{I}_2=\begin{bmatrix}1-4+3&6-6+0\\-2+2+0&-3+0+3\end{bmatrix}$
$\Rightarrow\text{A}^2-2\text{A}+3\text{I}_2=\begin{bmatrix}0&0\\0&0\end{bmatrix}=0$
Hence proved.

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