If A= [ ll 4 & 2 1 & 1 ] Find (A-2I)(A - 3I) - Vidyadip

Question

If $A=\left[\begin{array}{ll}4 & 2 \\ 1 & 1\end{array}\right]$ Find $(A-2I)(A - 3I)$

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Answer

$A-2 I=\left[\begin{array}{ll}4 & 2 \\ 1 & 1\end{array}\right]-2\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}4 & 2 \\ 1 & 1\end{array}\right]-\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]=\left[\begin{array}{cc}2 & 2 \\ 1 & -1\end{array}\right]$
$A-3 I=\left[\begin{array}{ll}4 & 2 \\ 1 & 1\end{array}\right]-3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}4 & 2 \\ 1 & 1\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ 1 & -2\end{array}\right]$
$\begin{array}{l}(A-2 I)(A-3 I) =\left[\begin{array}{cc}2 & 2 \\ 1 & -1\end{array}\right]\left[\begin{array}{cc}1 & 2 \\ 1 & -2\end{array}\right] \end{array} $
$ =\left[\begin{array}{ll}2+2 & 4-4 \\ 1-1 & 2+2\end{array}\right] $
$ =\left[\begin{array}{ll}4 & 0 \\ 0 & 4\end{array}\right]$

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