Chapter 3.6, Problem 30PPS

To determine

To verify: whether the equation $k(PQ)=P(kQ)$ is possible or not

Explanation of Solution

Given information: $P=\left[\begin{array}{cc}4& -1\\ 1& 2\end{array}\right],Q=\left[\begin{array}{cc}6& 4\\ -2& -5\end{array}\right],R=\left[\begin{array}{cc}4& 6\\ -6& 4\end{array}\right],k=2$

Proof:

First find $PQ$

  $PQ=\left[\begin{array}{cc}4& -1\\ 1& 2\end{array}\right]\left[\begin{array}{cc}6& 4\\ -2& -5\end{array}\right]=\left[\begin{array}{cc}24+2& 16+5\\ 6-4& 4-10\end{array}\right]=\left[\begin{array}{cc}26& 21\\ 2& -6\end{array}\right]$

Now find $k(PQ)$ for $k=2$

  $k(PQ)=2(PQ)=2\left[\begin{array}{cc}26& 21\\ 2& -6\end{array}\right]=\left[\begin{array}{cc}52& 42\\ 4& -12\end{array}\right](i)$

Now find $kQ$ for $k=2$

  $kQ=2\left[\begin{array}{cc}6& 4\\ -2& -5\end{array}\right]=\left[\begin{array}{cc}12& 8\\ -4& -10\end{array}\right]$

Then find $P(kQ)$

  $P(kQ)=\left[\begin{array}{cc}4& -1\\ 1& 2\end{array}\right]\left[\begin{array}{cc}12& 8\\ -4& -10\end{array}\right]=\left[\begin{array}{cc}48+4& 32+10\\ 12-8& 8-20\end{array}\right]=\left[\begin{array}{cc}52& 42\\ 4& -12\end{array}\right](ii)$

From $(i)$ and $(ii)$ ,

  $k(PQ)=P(kQ)$