Question

Given A = ${\displaystyle \left[\begin{array}{cc}p& 0\\ 0& 2\end{array}\right]}$, B = ${\displaystyle \left[\begin{array}{cc}0& -q\\ 1& 0\end{array}\right]}$, C = ${\displaystyle \left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]}$ and BA = C². Find the values of p and q.

Answer 1

Given: A = ${\displaystyle \left[\begin{array}{cc}p& 0\\ 0& 2\end{array}\right]}$, B = ${\displaystyle \left[\begin{array}{cc}0& -q\\ 1& 0\end{array}\right]}$, C = ${\displaystyle \left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]}$ and BA = C²

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

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

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

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

And C² = C × C

= ${\displaystyle \left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]\times \left[\begin{array}{cc}2& -2\\ 2& 2\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2\times 2+(-2)\times 2& 2\times (-2)+(-2)\times 2\\ 2\times 2+2\times 2& 2\times (-2)+2\times 2\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}4-4& -4-4\\ 4+4& -4+4\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}0& -8\\ 8& 0\end{array}\right]}$

∵ BA = C²

∴ ${\displaystyle \left[\begin{array}{cc}0& -2q\\ p& 0\end{array}\right]=\left[\begin{array}{cc}0& -8\\ 8& 0\end{array}\right]}$

Comparing the elements; –2q = –8

${\displaystyle \Rightarrow q=\frac{-8}{-2}=4}$

p = 8

Hence p = 8, q = 4