Let $\mathrm{A}=\left(\begin{array}{cc}4& -2\\ \mathrm{\alpha }& \mathrm{\beta }\end{array}\right)$

If ${\mathrm{A}}^2+\mathrm{\gamma A}+18 \mathrm{I}=0$, then det (A) is equal to ___________.

The characteristic equation for A is $\left|\mathrm{A}-\mathrm{\lambda I}\right|=0$

$\begin{array}{l}\Rightarrow \left|\begin{array}{cc}4-\mathrm{\lambda }& -2\\ \mathrm{\alpha }& \mathrm{\beta }-\mathrm{\lambda }\end{array}\right|=0\\ \Rightarrow (4-\mathrm{\lambda })(\mathrm{\beta }-\mathrm{\lambda })+2\mathrm{\alpha }=0\\ \Rightarrow {\mathrm{\lambda }}^2-(\mathrm{\beta }+4)\mathrm{\lambda }+4\mathrm{\beta }+2\mathrm{\alpha }=0\end{array}$

Put $\mathrm{\lambda }=\mathrm{A}$

${\mathrm{A}}^2-(\mathrm{\beta }+4)\mathrm{A}+(4\mathrm{\beta }+2\mathrm{\alpha })\mathrm{I}=0$

On comparison

$$\begin{gathered} -9(\mathrm{\beta }+4)=\mathrm{\gamma }\mathrm{\&}4\mathrm{\beta }+2\mathrm{\alpha }=18 \\ \mathrm{and} \left|\mathrm{A}\right|=4\mathrm{\beta }+2\mathrm{\alpha }=18 \end{gathered}$$