Question

Given $P=\left[\begin{array}{cc}p& 0\\ 0& 2\end{array}\right],Q=\left[\begin{array}{cc}0& -q\\ 1& 0\end{array}\right]\text{and}R=\left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]$

If $QP=R^2$, Find the values of p and q.

• A. p = 8, q = 4
• B. p = -8, q = -4
• C. p = 4, q = 8
• D. p = -4, q = -8

Answer 1

The correct option is A

p = 8, q = 4

$QP=\left[\begin{array}{cc}0& -q\\ 1& 0\end{array}\right]\left[\begin{array}{cc}p& 0\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}0+0& 0-2q\\ p+0& 0+0\end{array}\right]=\left[\begin{array}{cc}0& -2q\\ p& 0\end{array}\right]$

$R^2=\left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]\left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]=\left[\begin{array}{cc}4-4& -4-4\\ 4+4& -4+4\end{array}\right]=\left[\begin{array}{cc}0& -8\\ 8& 0\end{array}\right]$

Since $QP=R^2$, Equating the terms
$-2q=-8\Rightarrow q=4p=8$