To find the inverse of the given matrix, let's denote the matrix by [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
1 & 2 & 5 \\
3 & 5 & 9 \\
1 & 1 & -2
\end{pmatrix}
\][/tex]
We need to determine which of the provided choices correctly represents the inverse of [tex]\( A \)[/tex].
The options are:
A.
[tex]\[
\begin{pmatrix}
-19 & 9 & -7 \\
15 & -7 & 6 \\
-2 & 1 & -1
\end{pmatrix}
\][/tex]
B.
[tex]\[
\begin{pmatrix}
-19 & 9 & -7 \\
-2 & 1 & -1 \\
15 & -7 & 6
\end{pmatrix}
\][/tex]
C.
[tex]\[
\begin{pmatrix}
5 & -7 & 6 \\
-2 & 1 & -1
\end{pmatrix}
\][/tex]
D.
[tex]\[
\begin{pmatrix}
-19 & 9 \\
15 & -7 \\
\-2 & 1
\end{pmatrix}
\][/tex]
E. The matrix is non-invertible.
Given the matrix [tex]\( A \)[/tex] and the potential inverse matrices, we can verify through direct computation or analysis which matrix is indeed the inverse.
Upon correct calculations and comparison, the correct inverse of the given matrix [tex]\( A \)[/tex] is choice A:
[tex]\[
\begin{pmatrix}
-19 & 9 & -7 \\
15 & -7 & 6 \\
-2 & 1 & -1
\end{pmatrix}
\][/tex]
Thus, the answer is:
A.
[tex]\[
\begin{pmatrix}
-19 & 9 & -7 \\
15 & -7 & 6 \\
-2 & 1 & -1
\end{pmatrix}
\][/tex]
