If $\left[\begin{array}{cc}2\mathrm{\alpha }+1& 3\mathrm{\beta }\\ 0& {\mathrm{\beta }}^2-5\mathrm{\beta }\end{array}\right]=\left[\begin{array}{cc}\mathrm{\alpha }+3& {\mathrm{\beta }}^2+2\\ 0& -6\end{array}\right]$ find the value of $\mathrm{\alpha }$ + $\mathrm{\beta }$

The given matrices will be equal, iff

$\begin{array}{l}2\mathrm{\alpha }+1=\mathrm{\alpha }+3\Rightarrow \mathrm{\alpha }=2\\ 3\mathrm{\beta }={\mathrm{\beta }}^2+2\Rightarrow {\mathrm{\beta }}^2-3\mathrm{\beta }+2=0\\ \mathrm{\beta }=1,2\text{ and }{\mathrm{\beta }}^2-5\mathrm{\beta }=-6---\left(\mathrm{i}\right)\\ {\mathrm{\beta }}^2-5\mathrm{\beta }+6=0\\ \mathrm{\beta }=2,3----\left(\mathrm{ii}\right)\end{array}$

From Eqs. (i) and (i), we get  $\mathrm{\beta }$ =2

$\mathrm{\alpha }=2,\mathrm{\beta }=2$