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Compute $ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $

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  • The sum / difference $A(+/-)B$ of two $m$-by-$n$ matrices $A$ and $B$ is calculated entrywise: $(A (+/-) B)_{i,j} = A_{i,j} +/- B_{i,j}$ where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
$\begin{bmatrix}a & b\\-b& a\end{bmatrix}\begin{bmatrix}a &- b\\b& a\end{bmatrix}$
On multiplying we get,
$\begin{bmatrix}a^2+b^2 & -ab+ab\\-ab+ab & b^2+a^2\end{bmatrix}$
$\Rightarrow \begin{bmatrix}a^2+b^2 & 0\\0&a^2+b^2\end{bmatrix}$
$\Rightarrow a^2+b^2\begin{bmatrix}1 &0\\0 &1\end{bmatrix}$
$\Rightarrow a^2+b^2$
answered Sep 26, 2013 by sreemathi.v
 

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