To find the inverse of the matrix
[tex]\[
\begin{pmatrix}
1 & 1 & 1 \\
0 & 1 & 2 \\
1 & 0 & 1
\end{pmatrix},
\][/tex]
we can follow these steps:
1. Determine the determinant of the matrix: This is essential as it's a requirement for the matrix to be invertible. The determinant must be non-zero.
2. Find the matrix of minors: Calculate the determinant of the 2x2 minor matrices for each element in the original matrix.
3. Find the matrix of cofactors: Apply a checkerboard pattern of plus and minus signs to the matrix of minors.
4. Transpose the matrix of cofactors: This forms the adjugate (or adjoint) of the matrix.
5. Divide each element by the determinant: This will yield the inverse of the original matrix.
However, I will directly provide you with the result obtained from the inverse calculation:
The inverse of the matrix
[tex]\[
\begin{pmatrix}
1 & 1 & 1 \\
0 & 1 & 2 \\
1 & 0 & 1
\end{pmatrix}
\][/tex]
is
[tex]\[
\begin{pmatrix}
0.5 & -0.5 & 0.5 \\
1 & 0 & -1 \\
-0.5 & 0.5 & 0.5
\end{pmatrix}.
\][/tex]
