Compute the following products.

(i)

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

(ii)

$\left[ {\begin{array}{cccccccccccccccccccc}1\\2\\3\end{array}} \right][\begin{array}{cccccccccccccccccccc}2&3&4\end{array}]$

(iii)

$\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 2}\\2&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&2&3\\2&3&1\end{array}} \right]$

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

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

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

.:

(i)

We have, $\left[ {\begin{array}{cccccccccccccccccccc}a&b\\{ - b}&a\end{array}} \right]$ $\left[ {\begin{array}{cccccccccccccccccccc}a&{ - b}\\b&a\end{array}} \right]$

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

(ii) We have, $\left[ {\begin{array}{cccccccccccccccccccc}1\\2\\3\end{array}} \right][\begin{array}{cccccccccccccccccccc}2&3&4\end{array}] = \left[ {\begin{array}{cccccccccccccccccccc}{1 \times 2}&{1 \times 3}&{1 \times 4}\\{2 \times 2}&{2 \times 3}&{2 \times 4}\\{3 \times 2}&{3 \times 3}&{3 \times 4}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}2&3&4\\4&6&8\\6&9&{12}\end{array}} \right]$

(iii)

We have, $\left[ {\begin{array}{cccccccccccccccccccc}1&{ - 2}\\2&3\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}1&2&3\\2&3&1\end{array}} \right]$

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

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

(iv)

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

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

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

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

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

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

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

(vi)

$\left[ {\begin{array}{cccccccccccccccccccc}3&{ - 1}&3\\{ - 1}&0&2\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}2&{ - 3}\\1&0\\3&1\end{array}} \right]$

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

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