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