Question

Compute the indicated products:
(i) $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

(ii) $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ -3& 2& -1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$

Answer 1

$$\begin{gathered} \left(i\right)\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}a\times a+b\times b& a\times \left(-b\right)+b\times a\\ \left(-b\right)\times a+a\times b& \left(-b\right)\times \left(-b\right)+a\times a\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}{a}^2+b^2& -ab+ab\\ -ab+ab& b^2+a^2\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}{a}^2+b^2& 0\\ 0& a^2+b^2\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(ii\right)\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ -3& 2& -1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1\times 1+\left(-2\right)\times \left(-3\right)& 1\times 2+\left(-2\right)\times 2& 1\times 3+\left(-2\right)\times \left(-1\right)\\ 2\times 1+3\times \left(-3\right)& 2\times 2+3\times 2& 2\times 3+3\times \left(-1\right)\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1+6& 2-4& 3+2\\ 2-9& 4+6& 6-3\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}7& -2& 5\\ -7& 10& 3\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(iii\right)\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}2\times 1+3\times 0+4\times 3& 2\times \left(-3\right)+3\times 2+4\times 0& 2\times 5+3\times 4+4\times 5\\ 3\times 1+4\times 0+5\times 3& 3\times \left(-3\right)+4\times 2+5\times 0& 3\times 5+4\times 4+5\times 5\\ 4\times 1+5\times 0+6\times 3& 4\times \left(-3\right)+5\times 2+6\times 0& 4\times 5+5\times 4+6\times 5\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}2+0+12& -6+6+0& 10+12+20\\ 3+0+15& -9+8+0& 15+16+25\\ 4+0+18& -12+10+0& 20+20+30\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}14& 0& 42\\ 18& -1& 56\\ 22& -2& 70\end{array}\right] \end{gathered}$$