Let A = ( cases 4 & - 2 α & β cases ) . If A ^ 2 + + 18I = O , then...

Given A = \left[\begin{cases}4 & - 2 \\ \alpha & \beta\end{cases}\right] ⇒ det\left(A\right) = 4\beta + 2\alpha

Characteristic equation of the matrix is

\left[$\begin{cases}4 - \lambda & - 2 \\ \alpha & \beta - \lambda\end{cases}$\right] = 0$</p><p>$⇒ 4\beta + {\lambda}^{2} - \left(\beta + 4\right)\lambda + 2\alpha = 0$</p><p>$⇒ {\lambda}^{2} - \left(\beta + 4\right)\lambda + 2\alpha + 4\beta = 0$</p><p>Comparing with given equation${A}^{2} + \gammaA + 18I = 0$, we get</p><p>$⇒ \gamma = - \left(\beta + 4\right) \text{$and$} 2\alpha + 4\beta = 18$</p><p>Hence$det \left(A\right) = 4\beta + 2\alpha = 18