Question

The product $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$ is equal to

• A. $\left[\begin{array}{cc}{a}^2+b^2& 0\\ 0& a^2+b^2\end{array}\right]$
• B. $\left[\begin{array}{cc}(a+b{)}^2& 0\\ (a+b{)}^2& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}{a}^2+b^2& 0\\ a^2+b^2& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$

Answer 1

Answer: (A)

$\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$
$=\left[\begin{array}{cc}a\times a+b\times b& a\times (-b)+b\times a\\ (-b)\times a+a\times b& (-b)\times (-b)+a\times a\end{array}\right]$
$=\left[\begin{array}{cc}{a}^2+b^2& -ab+ab\\ -ab+ab& b^2+a^2\end{array}\right]=\left[\begin{array}{cc}{a}^2+b^2& 0\\ 0& b^2+a^2\end{array}\right]$