To find the inverse of matrix [tex]\( A \)[/tex], denoted [tex]\( A^{-1} \)[/tex], we follow these steps:
1. Write down the matrix [tex]\( A \)[/tex]:
[tex]\[
A = \left[\begin{array}{ccc}
9 & 2 & 1 \\
5 & -1 & 6 \\
4 & 0 & -2
\end{array}\right]
\][/tex]
2. Check if [tex]\( A \)[/tex] is invertible:
- A matrix is invertible if and only if its determinant is non-zero.
- Calculate the determinant of [tex]\( A \)[/tex]. If the determinant is zero, the matrix is not invertible.
3. Calculate the cofactors, adjugate, and determinant:
- Each element of the inverse matrix [tex]\( A^{-1} \)[/tex] is derived from the cofactors of [tex]\( A \)[/tex].
Given the detailed calculation steps are purposefully not shown here, the final inverse matrix [tex]\( A^{-1} \)[/tex] is derived correctly as:
[tex]\[
A^{-1} = \left[\begin{array}{ccc}
0.02222222222222223 & 0.044444444444444425 & 0.14444444444444446 \\
0.37777777777777777 & -0.24444444444444438 & -0.5444444444444445 \\
0.044444444444444446 & 0.08888888888888889 & -0.21111111111111114
\end{array}\right]
\][/tex]
So, the inverse of matrix [tex]\( A \)[/tex] is:
[tex]\[
A^{-1} = \left[\begin{array}{ccc}
0.02222222222222223 & 0.044444444444444425 & 0.14444444444444446 \\
0.37777777777777777 & -0.24444444444444438 & -0.5444444444444445 \\
0.044444444444444446 & 0.08888888888888889 & -0.21111111111111114
\end{array}\right]
\][/tex]
