Let $A = \left[ {\begin{array}{cccccccccccccccccccc}2&4\\3&2\end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc}1&3\\{ - 2}&5\end{array}} \right],C = \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&5\\3&4\end{array}} \right]$

Find each of the following :

(i) A+ B

(ii) A$-$B

(iii) 3A$-$ C

(iv) AB

(v) BA

.:

Here, A is a 2 × 2 matrix, B is a 2 × 2 matrix and C is a 2 × 2 matrix. So, A, B, C are comparable.

(i)

$A + B = \left[ {\begin{array}{cccccccccccccccccccc}2&4\\3&2\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}1&3\\2&5\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{2 + 1}&{4 + 3}\\{3 + ( - 2)}&{2 + 5}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}3&7\\1&7\end{array}} \right]$

(ii)

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

(iii)

$3A - C = 3\left[ {\begin{array}{cccccccccccccccccccc}2&4\\3&2\end{array}} \right] - \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&5\\3&4\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}6&{12}\\9&6\end{array}} \right] - \left[ {\begin{array}{cccccccccccccccccccc}{ - 2}&5\\3&4\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}{6 + 2}&{12 - 5}\\{9 - 3}&{6 - 4}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}8&7\\6&2\end{array}} \right]$

(iv)

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

$= \left[ {\begin{array}{cccccccccccccccccccc}{2 - 8}&{6 + 20}\\{3 - 4}&{9 + 10}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{ - 6}&{26}\\{ - 1}&{19}\end{array}} \right]$

(v)

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

$= \left[ {\begin{array}{cccccccccccccccccccc}{2 + 9}&{4 + 6}\\{ - 4 + 15}&{ - 8 + 10}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{11}&{10}\\{11}&2\end{array}} \right]$