Question

Let $A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$ and $\left(\right.10\left.\right)B=\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{bmatrix}$ , If $B$ is the inverse of matrix $A$ , then $\alpha $ is

• A. $5$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

Answer: (A)

$A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$
Cofactors of various entries are
$4,-5,1;2,0,-2;2,5,3$
$\left|\right.A\left|\right.=1\times 4+\left(\right.-1\left.\right)\times -5+1\times 1=10$
Cofactor Matrix $C=\begin{bmatrix} 4 & -5 & 1 \\ 2 & 0 & -2 \\ 2 & 5 & 3 \end{bmatrix}$
$\therefore AdjA=C^{T}=\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$
$\therefore A^{- 1}=\frac{\text{ Adj} \textit{A} \, }{\left|\right. A \left|\right.}=\frac{1}{10}\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$
Comparing, we get $\alpha =5$