Question

The inverse of the matrix $\left[\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]$

• A. $\left[\begin{array}{ccc}1& 1& 1\\ 3& 4& 3\\ 3& 3& 4\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 3& 1\\ 4& 3& 8\\ 3& 4& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& 4\\ 3& 4& 3\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$

Answer 1

Answer: (D)

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