Question
If $A=\left[\begin{array}{ll}a & 0 \\ 0 & 2\end{array}\right], B =\left[\begin{array}{cc}0 & -b \\ 1 & 0\end{array}\right], M=\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right]$ and $B A=M^2$ find the values of $a$ and $b$.
✓
Answer
$\begin{array}{l}B A=\left[\begin{array}{cc}0 & -b \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}a & 0 \\ 0 & 2\end{array}\right] \end{array} $
$ =\left[\begin{array}{cc}0+0 & 0-2 b \\ a+0 & 0+0\end{array}\right] $
$ =\left[\begin{array}{cc}0 & -2 b \\ a & 0\end{array}\right]$
$\begin{array}{l}M^2=\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right]\end{array} $
$ =\left[\begin{array}{cc}1-1 & -1-1 \\ 1+1 & -1+1\end{array}\right] $
$ =\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$
Given $B A=M^2$
$\left[\begin{array}{cc}0 & -2 b \\ a & 0\end{array}\right]=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$
Comparing the corresponding elements we get
$a =2$
$-2 b=-2 $
$\Rightarrow b=1$
$ =\left[\begin{array}{cc}0+0 & 0-2 b \\ a+0 & 0+0\end{array}\right] $
$ =\left[\begin{array}{cc}0 & -2 b \\ a & 0\end{array}\right]$
$\begin{array}{l}M^2=\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ 1 & 1\end{array}\right]\end{array} $
$ =\left[\begin{array}{cc}1-1 & -1-1 \\ 1+1 & -1+1\end{array}\right] $
$ =\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$
Given $B A=M^2$
$\left[\begin{array}{cc}0 & -2 b \\ a & 0\end{array}\right]=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$
Comparing the corresponding elements we get
$a =2$
$-2 b=-2 $
$\Rightarrow b=1$
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