Question
If $P=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|$ and $Q=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|$ find $P(Q P)$.
✓
Answer
$\begin{array}{l}P=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|_{2 \times 2}, Q=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|_{2 \times 2} \end{array} $
$ QP =\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right| $
$=\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|_{2 \times 2} $
$ P(Q P)=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|\left|\begin{array}{cc}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}{ll}14 & 13 \\ 13 & 14\end{array}\right|_{2 \times 2} $
$ QP =\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right| $
$=\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|_{2 \times 2} $
$ P(Q P)=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|\left|\begin{array}{cc}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}{ll}14 & 13 \\ 13 & 14\end{array}\right|_{2 \times 2} $
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