Let [A=([gathered 4 -2 gathered])].

Question

Let $$A=\left(\[\begin{gathered} 4 \quad  -2 \ \alpha \quad  \beta \end{gathered}$$\right)\].

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

Accepted Answer

Characteristic equation of A is given by

$$\left| {A - \lambda I} \right| = 0$$

$$\left| {\begin{pmatrix} {4 - \lambda } & { - 2}  \
   \alpha  & {\beta  - \lambda }  \

\end{pmatrix} } \right| = 0$$

$$ \Rightarrow {\lambda ^2} - (4 + \beta )\lambda  + (4\beta  + 2\alpha ) = 0$$

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

$$|A| = 4\beta  + 2\alpha  = 18$$

Solutions · 1