Question
If $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$ Find $A(B A)$
✓
Answer
$A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
$\begin{array}{l}B A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] \end{array} $
$ =\left[\begin{array}{ll}2+2 & 4+1 \\ 1+4 & 2+2\end{array}\right] $
$ =\left[\begin{array}{ll}4 & 5 \\ 5 & 4\end{array}\right]$
$\begin{array}{l}A(B A)=\left[\begin{array}{ll}1 & 2 \\ 4 & 5\end{array}\right],\left[\begin{array}{ll}4 & 5 \\ 5 & 4\end{array}\right] \end{array} $
$ =\left[\begin{array}{cc}4+10 & 5+8 \\ 8+5 & 10+4\end{array}\right] $
$ =\left[\begin{array}{ll}14 & 13 \\ 13 & 14\end{array}\right]$
$\begin{array}{l}B A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] \end{array} $
$ =\left[\begin{array}{ll}2+2 & 4+1 \\ 1+4 & 2+2\end{array}\right] $
$ =\left[\begin{array}{ll}4 & 5 \\ 5 & 4\end{array}\right]$
$\begin{array}{l}A(B A)=\left[\begin{array}{ll}1 & 2 \\ 4 & 5\end{array}\right],\left[\begin{array}{ll}4 & 5 \\ 5 & 4\end{array}\right] \end{array} $
$ =\left[\begin{array}{cc}4+10 & 5+8 \\ 8+5 & 10+4\end{array}\right] $
$ =\left[\begin{array}{ll}14 & 13 \\ 13 & 14\end{array}\right]$
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