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