Compute the indicated products

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

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

$=\left[\begin{array}{lr}a(a)+b(b) & a(-b)+b(a) \\ -b(a)+a(b) & -b(-b)+a(a)\end{array}\right]$

$=\left[\begin{array}{ll}a^{2}+b^{2} & -a b+a b \\ -a b+a b & b^{2}+a^{2}\end{array}\right]=\left[\begin{array}{lc}a^{2}+b^{2} & 0 \\ 0 & a^{2}+b^{2}\end{array}\right]$\

(ii) $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\left[\begin{array}{lll}2 & 3 & 4\end{array}\right]=\left[\begin{array}{lll}1(2) & 1(3) & 1(4) \\ 2(2) & 2(3) & 2(4) \\ 3(2) & 3(3) & 3(4)\end{array}\right]=\left[\begin{array}{lll}2 & 3 & 4 \\ 4 & 6 & 8 \\ 6 & 9 & 12\end{array}\right]$

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

$=\left[\begin{array}{lll}1(1)-2(2) & 1(2)-2(3) & 1(3)-2(1) \\ 2(1)+3(2) & 2(2)+3(3) & 2(3)+3(1)\end{array}\right]$

$=\left[\begin{array}{lll}1-4 & 2-6 & 3-2 \\ 2+6 & 4+9 & 6+3\end{array}\right]=\left[\begin{array}{rcr}-3 & -4 & 1 \\ 8 & 13 & 9\end{array}\right]$

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

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

$=\left[\begin{array}{lll}2+0+12 & -6+6+0 & 10+12+20 \\ 3+0+15 & -9+8+0 & 15+16+25 \\ 4+0+18 & -12+10+0 & 20+20+30\end{array}\right]=\left[\begin{array}{ccr}14 & 0 & 42 \\ 18 & -1 & 56 \\ 22 & -2 & 70\end{array}\right]$

(v) $\left[\begin{array}{rr}2 & 1 \\ 3 & 2 \\ -1 & 1\end{array}\right]\left[\begin{array}{rrr}1 & 0 & 1 \\ -1 & 2 & 1\end{array}\right]$

$=\left[\begin{array}{lll}2(1)+1(-1) & 2(0)+1(2) & 2(1)+1(1) \\ 3(1)+2(-1) & 3(0)+2(2) & 3(1)+2(1) \\ -1(1)+1(-1) & -1(0)+1(2) & -1(1)+1(1)\end{array}\right]$

$=\left[\begin{array}{lll}2-1 & 0+2 & 2+1 \\ 3-2 & 0+4 & 3+2 \\ -1-1 & 0+2 & -1+1\end{array}\right]=\left[\begin{array}{ccr}1 & 2 & 3 \\ 1 & 4 & 5 \\ -2 & 2 & 0\end{array}\right]$

(vi) $\left[\begin{array}{rrr}3 & -1 & 3 \\ -1 & 0 & 2\end{array}\right]\left[\begin{array}{rr}2 & -3 \\ 1 & 0 \\ 3 & 1\end{array}\right]$

$=\left[\begin{array}{cc}3(2)-1(1)+3(3) & 3(-3)-1(0)+3(1) \\ -1(2)+0(1)+2(3) & -1(-3)+0(0)+2(1)\end{array}\right]$

$=\left[\begin{array}{cc}6-1+9 & -9-0+3 \\ -2+0+6 & 3+0+2\end{array}\right]=\left[\begin{array}{cc}14 & -6 \\ 4 & 5\end{array}\right]$