Question

Compute the indicated product
$\left(i\right)$ $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$
$\left(ii\right)$ $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$
$\left(iii\right)$ $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$
$\left(iv\right)$ $\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$
$\left(v\right)$ $\left[\begin{array}{cc}2& 1\\ 3& 2\\ -1& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]$
$\left(vi\right)$ $\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$

Answer 1

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

$\left(ii\right)$ Given: $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$
$={\left[\begin{array}{ccc}1\times 2& 1\times 3& 1\times 4\\ 2\times 2& 2\times 3& 2\times 4\\ 3\times 2& 3\times 3& 3\times 4\end{array}\right]}_{3\times 3}$
$=\left[\begin{array}{ccc}2& 3& 4\\ 4& 6& 8\\ 6& 9& 12\end{array}\right]$

$\left(iii\right)$ Given: ${\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]}_{2\times 2}{\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]}_{2\times 3}$
$=\left[\begin{array}{ccc}1\times 1+(-2)\times 2& 1\times 2+(-2)\times 3& 1\times 3+(-2)\times 1\\ 2\times 1+3\times 2& 2\times 2+3\times 3& 2\times 3+3\times 1\end{array}\right]$
$=\left[\begin{array}{ccc}1-4& 2-6& 3-2\\ 2+6& 4+9& 6+3\end{array}\right]$
$=\left[\begin{array}{ccc}-3& -4& 1\\ 8& 13& 9\end{array}\right]$

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

$\left(v\right)$ Given: ${\left[\begin{array}{cc}2& 1\\ 3& 2\\ -1& 1\end{array}\right]}_{3\times 2}{\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]}_{2\times 3}$
$={\left[\begin{array}{ccc}2\times 1+1\times (-1)& 2\times 0+1\times 2& 2\times 1+1\times 1\\ 3\times 1+2\times (-1)& 3\times 0+2\times 2& 3\times 1+2\times 1\\ -1\times 1+1\times (-1)& -1\times 0+1\times 2& -1\times 1+1\times 1\end{array}\right]}_{3\times 3}$
$=\left[\begin{array}{ccc}2-1& 0+2& 2+1\\ 3-2& 0+4& 3+2\\ -1-1& 0+2& -1+1\end{array}\right]$
$=\left[\begin{array}{ccc}1& 2& 3\\ 1& 4& 5\\ -2& 2& 0\end{array}\right]$

$\left(vi\right)$ Given: ${\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]}_{2\times 3}{\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]}_{3\times 2}$
$=\left[\begin{array}{cc}3\left(2\right)-1\left(1\right)+3\left(3\right)& 3(-3)-1\left(0\right)+3\left(1\right)\\ -1\left(2\right)+0\left(1\right)+2\left(3\right)& -1(-3)+0\left(0\right)+2\left(1\right)\end{array}\right]$
$=\left[\begin{array}{cc}6-1+9& -9-0+3\\ -2+0+6& 3+0+2\end{array}\right]$
$=\left[\begin{array}{cc}14& -6\\ 4& 5\end{array}\right]$