Question

If $A=\begin{bmatrix} 2 & 2 \\ 9 & 4 \end{bmatrix}$ and $A^{2}+aA+bΙ=O$ , then $a+2b$ is equal to (where, $Ι$ is an identity matrix and $O$ is a null matrix of order $2$ respectively)

• A. $27$
• B. $-26$
• C. $24$
• D. $12$

Answer 1

Answer: (B)

$A=\begin{bmatrix} 2 & 2 \\ 9 & 4 \end{bmatrix}\Rightarrow A^{2}=\begin{bmatrix} 2 & 2 \\ 9 & 4 \end{bmatrix}\begin{bmatrix} 2 & 2 \\ 9 & 4 \end{bmatrix}=\begin{bmatrix} 22 & 12 \\ 54 & 34 \end{bmatrix}$
$aA=\begin{bmatrix} 2a & 2a \\ 9a & 4a \end{bmatrix},bΙ=\begin{bmatrix} b & 0 \\ 0 & b \end{bmatrix}$
$A^{2}+aA+bΙ=\begin{bmatrix} 22+2a+b & 12+2a \\ 9a+54 & 34+4a+b \end{bmatrix}=O$
$\Rightarrow a=-6$
$\Rightarrow 22+2a+b=0\Rightarrow b=-10$
$\Rightarrow a+2b=-26$