Question:
If $A=\left[a_{i j}\right]=\left[\begin{array}{rrr}2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2\end{array}\right]$ and $B=\left[b_{i j}\right]=$ $\left[\begin{array}{cc}2 & -1 \\ -3 & 4 \\ 1 & 2\end{array}\right]$
then find (i) $a_{22}+b_{21}$
(ii) $a_{11} b_{11}+a_{22} b_{22}$
Solution:
(i)
$a_{22}+b_{21}$
Here,
$a_{22}=4$ and $b_{21}=-3$
$\Rightarrow a_{22}+b_{21}=4-3=1$
(ii)
$a_{11} b_{11}+a_{22} b_{22}$
Here,
$a_{11}=2, b_{11}=2, a_{22}=4$ and $b_{22}=4$
$\Rightarrow a_{11} b_{11}+a_{22} b_{22}=2 \times 2+4 \times 4$
$\Rightarrow a_{11} b_{11}+a_{22} b_{22}=4+16$
$\Rightarrow a_{11} b_{11}+a_{22} b_{22}=20$