Find the values of $a$ and $b$, if $A = B$, where
$A = \left[ {\begin{array}{cccccccccccccccccccc}{a + 4}&{3b}\\8&{ - 6}\end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc}{2a + 2}&{{b^2} + 2}\\8&{{b^2} - 5b}\end{array}} \right]$

We have, $A = {\left[ {\begin{array}{cccccccccccccccccccc}{a + 4}&{3b}\\8&{ - 6}\end{array}} \right]_{2 \times 2}}$ and $B = {\left[ {\begin{array}{cccccccccccccccccccc}{2a + 2}&{{b^2} + 2}\\8&{{b^2} - 5b}\end{array}} \right]_{2 \times 2}}$

Hence , $A = B$
By equality of matrices As we know, each element of A is equal to the corresponding element of B,

that is ${a_{ij}} = {b_{ij}}$ for all $i$ and $j$.
$\therefore$ ${a_{11}} = {b_{11}} \Rightarrow a + 4 = 2a + 2 \Rightarrow a = 2$
${a_{12}} = {b_{12}} \Rightarrow 3b = {b^2} + 2 \Rightarrow {b^2} = 3b - 2$

and ${a_{22}} = {b_{22}} \Rightarrow - 6 = {b^2} - 5b$
$\Rightarrow$ $- 6 = 3b - 2 - 5b$

$\Rightarrow$ $2b = 4 \Rightarrow b = 2$
$\therefore$ $a = 2$ and $b = 2$