The inverse of the matrix $\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 &…
The inverse of the matrix $\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ is
- $\left[\begin{array}{lll}1 & 1 & 1 \\ 3 & 4 & 3 \\ 3 & 3 & 4\end{array}\right]$
- $\left[\begin{array}{lll}1 & 3 & 1 \\ 4 & 3 & 8 \\ 3 & 4 & 1\end{array}\right]$
- $\left[\begin{array}{lll}1 & 1 & 1 \\ 3 & 3 & 4 \\ 3 & 4 & 3\end{array}\right]$
- $\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$
Solution
Let
$
A=\left[\begin{array}{ccc}
7 & -3 & -3 \\
-1 & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
$
Now,
$
\begin{aligned}
|A| & =7(1-0)+3(-1-0)-3(0+1) \\
& =1
\end{aligned}
$
Cofactors of matrix $A$ are
$
\begin{aligned}
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 \quad \operatorname{adj}(A) & =\left[\begin{array}{lll}
1 & 1 & 1 \\
3 & 4 & 3 \\
3 & 3 & 4
\end{array}\right]^T \\
& =\left[\begin{array}{lll}
1 & 3 & 3 \\
1 & 4 & 3 \\
1 & 3 & 4
\end{array}\right] \\
\therefore \quad A^{-1} & =\frac{\operatorname{adj}(A)}{|A|}=\left[\begin{array}{lll}
1 & 3 & 3 \\
1 & 4 & 3 \\
1 & 3 & 4
\end{array}\right]
\end{aligned}
$
Asked in: AP EAMCET 2008
Practice more Matrices questions on Aicharya