Question

If $A = \begin{bmatrix}2&-3\\ -4&1\end{bmatrix} $, then adj $\left(3A^{2} + 12 A\right)$ is equal to :

• A. $\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix} $
• B. $\begin{bmatrix} 51 &84 \\ 63 & 72 \end{bmatrix} $
• C. $\begin{bmatrix} 72& -63 \\ -84 & 51\end{bmatrix} $
• D. $\begin{bmatrix} 72& -84 \\ -63 & 51\end{bmatrix} $

Answer 1

Answer: (A)

$A = \begin{bmatrix}2&-3\\ -4&1\end{bmatrix}$
$\left|A-\lambda l\right| = \begin{bmatrix}2-\lambda&-3\\ -4&1-\lambda\end{bmatrix}$
$= \left(2-2\lambda-\lambda+\lambda^{2}\right) - 12$
$f\left(\lambda\right) = \lambda^{2}-3\lambda-10$
$\because$ A satisfies $f \left(\lambda\right)$
$\therefore A^{2} - 3A -10l = 0$
$A^{2} - 3A = 10l$
$3A^{2} - 9A = 30l$
$3A^{2} + 12A = 30l + 21A$
$=\begin{bmatrix}30&0\\ 0&30\end{bmatrix}+\begin{bmatrix}42&-63\\ -84&21\end{bmatrix}$
$=\begin{bmatrix}72&-63\\ -84&51\end{bmatrix}$
$adj\left(3A^{2}+12A\right) = \begin{bmatrix}51&63\\ 84&72\end{bmatrix}$