Find the inverse of the matrix $A$ using minors, cofactors and adjugate.
\[
A=\left[\begin{array}{ccc}
0 & 2 & 2 \\
6 & 4 & -1 \\
1 & -1 & -1
\end{array}\right]
\]
Answer
\(\boxed{A^{-1} = \left(\begin{array}{lll} 0.5 & 0.25 & -0.67 \\ 0.25 & 0.17 & -0.33 \\ -1.17 & -0.58 & 1.17 \end{array}\right)}\)
Step 1 ：Find the determinant of the matrix \(A\): \[\text{det}(A) = -12\]
Step 2 ：Since the determinant is non-zero, the inverse of \(A\) exists.
Step 3 ：Find the adjugate of the matrix \(A\): \[\text{adj}(A) = \left(\begin{array}{lll} -6 & -3 & 8 \\ -3 & -2 & 4 \\ 14 & 7 & -14 \end{array}\right)\]
Step 4 ：Calculate the inverse of \(A\) using the formula: \[A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)\]
Step 5 ：\(\boxed{A^{-1} = \left(\begin{array}{lll} 0.5 & 0.25 & -0.67 \\ 0.25 & 0.17 & -0.33 \\ -1.17 & -0.58 & 1.17 \end{array}\right)}\)
