Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find $A\cdot A^{\prime }$.

Answer 1

Given : $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

$⟹$ $A^{{}^{\prime }}=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

Consider, $AA{}^{\prime }=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

$⟹$ $A=\left[\begin{array}{cc}1+4& 3+8\\ 3+8& 9+16\end{array}\right]$

$\therefore$ $AA{}^{\prime }=\left[\begin{array}{cc}5& 11\\ 11& 25\end{array}\right]$