Question

The inverse of the matrix $\begin{bmatrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix}$

• A. $\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 3 \\ 3 & 3 & 4\end{bmatrix}$
• B. $\begin{bmatrix}1 & 3 & 1 \\ 4 & 3 & 8 \\ 3 & 4 & 1\end{bmatrix}$
• C. $\begin{bmatrix}1 & 1 & 1 \\ 3 & 3 & 4 \\ 3 & 4 & 3\end{bmatrix}$
• D. $\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$

Answer 1

Answer: (D)

$A=\begin{bmatrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix}$
Now, $|A|=7(1-0)+3(-1-0)-3(0+1)$
$=1$
Cofactors of matrix $A$ are
$C_{11}=1, C_{12}=1, C_{13}=1 $
$C_{21}=3, C_{22}=4, C_{23}=3 $
$C_{31}=3, C_{32}=3, C_{33}=4$
$\therefore \text{adj}(A)=\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 3 \\ 3 & 3 & 4\end{bmatrix}^{T}$
$=\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$
$\therefore A^{-1}=\frac{\text{adj}(A)}{|A|}=\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}$