If f(x)=x^2-2 x-3 then find f(A) when A= [ ll 1 & 2 2 & 1 ] - Vidyadip

Question

If $f(x)=x^2-2 x-3$ then find $f(A)$ when $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$

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Answer

$f(A)=A^2-2 A-3 l $
$ A^2=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] \left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] $
$ =\left[\begin{array}{ll}1+4 & 2+2 \\ 2+2 & 4+1\end{array}\right] $
$ =\left[\begin{array}{ll}5 & 4 \\ 4 & 5\end{array}\right]$
$ \therefore f (A)=\left[\begin{array}{ll}5 & 4 \\ 4 & 5\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 4 & 2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] $
$ =\left[\begin{array}{ll}5-2-3 & 4-4-0 \\ 4-4-0 & 5-2-3\end{array}\right] $
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

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