Question

Let $A=\begin{bmatrix} -1 & \, \, \, 2 & -3 \\ -2 & \, \, \, 0 & \, \, \, 3 \\ \, \, \, 3 & -3 & \, \, \, 1 \end{bmatrix}$ be a matrix, then $\left|A\right|adj \, \left(A^{- 1}\right)$ is equal to

• A. $O_{3 \times 3}$
• B. $\begin{bmatrix} -1 & \, \, \, 2 & -3 \\ -2 & \, \, \, 0 & \, \, \, 3 \\ \, \, \, 3 & -3 & \, \, \, 1 \end{bmatrix}$
• C. $I_{3}$
• D. $\begin{bmatrix} -3 & -3 &    1 \\    3 &    0 & -2 \\ -1 &    2 & -3 \end{bmatrix}$

Answer 1

Answer: (B)

We know that,
$A^{- 1}adj \, \left(A^{- 1}\right)= \, \left|A^{- 1}\right|I_{3}$
$\Rightarrow A. \, A^{- 1}adj \, \left(A^{- 1}\right)= \, \left|A^{- 1}\right|\left(A I\right)_{3}$
$\Rightarrow \, adj \, \left(A^{- 1}\right)= \, \left|A^{- 1}\right|A$
$\Rightarrow \, \left|A\right|adj \, \left(A^{- 1}\right)= \, A$