Find the matrix A^2, where A= [a_ i j ] is a 2 2 matrix whose ele… - Vidyadip

Question

Find the matrix $A^2$, where $A=\left[a_{i j}\right]$ is a $2 \times 2$ matrix whose elements are given by $a_{i j}=$ maximum $(i, j)-$ minimum $(i, j)$ :

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Answer

Let $A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right] $
$a_{11}=\max (1,1)-\min (1,1)=1-1=0$
$a_{12}=\max (1,2)-\min (1,2)=2-1=1$
$a_{21}=\max (2,1)-\min (2,1)=2-1=1$
$a_{22}=\max (2,2)-\min (2,2)=2-2=0 $
$ \therefore A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
$ A^2=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

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