Question

If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, find $AB$ and $BA.$ Prove that $AB\ne BA$.

Answer 1

Given: $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
Now,
$AB=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}1\left(0\right)+0\left(1\right)& 1\left(1\right)+0\left(0\right)\\ 0\left(0\right)+(-1)\left(1\right)& 0\left(1\right)+(-1)\left(0\right)\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}0+0& 1+0\\ 0-1& 0+0\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\cdots \left(1\right)$

$BA=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
$\Rightarrow BA=\left[\begin{array}{cc}0\left(1\right)+1\left(0\right)& 0\left(0\right)+1(-1)\\ 1\left(1\right)+0\left(0\right)& 1\left(0\right)+0(-1)\end{array}\right]$
$\Rightarrow BA=\left[\begin{array}{cc}0+0& 0-1\\ 1+0& 0+0\end{array}\right]$
$\Rightarrow BA=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\cdots \left(2\right)$
From equation $\left(1\right)$ and $\left(2\right)$
$BA\ne AB$.
Hence proved.