Compute the indicated product [ cc a & b -b & a ] [ cc a & -b b &… - Vidyadip

Question

Compute the indicated product
$\left[\begin{array}{cc} {a} & {b} \\ {-b} & {a} \end{array}\right]\left[\begin{array}{cc} {a} & {-b} \\ {b} & {a} \end{array}\right]$

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Answer

According to the Question,
$\left[\begin{array}{cc} {a} & {b} \\ {-b} & {a} \end{array}\right]\left[\begin{array}{cc} {a} & {-b} \\ {b} & {a} \end{array}\right]$
= $\left[\begin{array}{cc} {a(a)+b(b)} & {a(-b)+b(a)} \\ {-b(a)+a(b)} & {-b(-b)+a(a)} \end{array}\right]$
= $\left[\begin{array}{cc} {a^{2}+b^{2}} & {-a b+a b} \\ {-a b+a b} & {b^{2}+a^{2}} \end{array}\right]$
= $\left[\begin{array}{cc} {a^{2}+b^{2}} & {0} \\ {0} & {b^{2}+a^{2}} \end{array}\right]$

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