Compute the indicated products:
Question:

Compute the indicated products:

(i) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]$

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

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

Solution:

(i) $\left[\begin{array}{cc}a & b \\ -b & a\end{array}\right]\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}a \times a+b \times b & a \times(-b)+b \times a \\ (-b) \times a+a \times b & (-b) \times(-b)+a \times a\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}a^{2}+b^{2} & -a b+a b \\ -a b+a b & b^{2}+a^{2}\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}a^{2}+b^{2} & 0 \\ 0 & a^{2}+b^{2}\end{array}\right]$

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

$\Rightarrow\left[\begin{array}{ccc}1 \times 1+(-2) \times(-3) & 1 \times 2+(-2) \times 2 & 1 \times 3+(-2) \times(-1) \\ 2 \times 1+3 \times(-3) & 2 \times 2+3 \times 2 & 2 \times 3+3 \times(-1)\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}1+6 & 2-4 & 3+2 \\ 2-9 & 4+6 & 6-3\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}7 & -2 & 5 \\ -7 & 10 & 3\end{array}\right]$

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

$\Rightarrow\left[\begin{array}{lll}2 \times 1+3 \times 0+4 \times 3 & 2 \times(-3)+3 \times 2+4 \times 0 & 2 \times 5+3 \times 4+4 \times 5 \\ 3 \times 1+4 \times 0+5 \times 3 & 3 \times(-3)+4 \times 2+5 \times 0 & 3 \times 5+4 \times 4+5 \times 5 \\ 4 \times 1+5 \times 0+6 \times 3 & 4 \times(-3)+5 \times 2+6 \times 0 & 4 \times 5+5 \times 4+6 \times 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{lll}2+0+12 & -6+6+0 & 10+12+20 \\ 3+0+15 & -9+8+0 & 15+16+25 \\ 4+0+18 & -12+10+0 & 20+20+30\end{array}\right]$

$\Rightarrow\left[\begin{array}{llr}14 & 0 & 42 \\ 18 & -1 & 56 \\ 22 & -2 & 70\end{array}\right]$