To determine the inverse of the given matrix
[tex]\[
\left[\begin{array}{ccc}
1 & 2 & 5 \\
3 & 6 & 9 \\
1 & 1 & -2
\end{array}\right]
,
\][/tex]
we assess its entries and calculate its inverse if it exists. Performing this calculation, we obtain the inverse matrix as:
[tex]\[
\left[\begin{array}{ccc}
\frac{7}{2} & -\frac{3}{2} & 2 \\
-\frac{5}{2} & \frac{7}{6} & -1 \\
\frac{1}{2} & -\frac{1}{6} & 0
\end{array}\right]
.
\][/tex]
Let's compare this result with the given answer choices:
A. [tex]\(\left[\begin{array}{ccc}-19 & 9 & -7 \\ 15 & -7 & 6 \\ -2 & 1 & -1\end{array}\right]\)[/tex]
B. [tex]\(\left[\begin{array}{ccc}-19 & 9 & -7 \\ -2 & 1 & -1 \\ 15 & -7 & 6\end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{ccc}5 & -7 & 6 \\ -2 & 1 & -1\end{array}\right]\)[/tex]
D. [tex]\(\left[\begin{array}{cc}-19 & 9 \\ 15 & -7 \\ -2 & 1\end{array}\right]\)[/tex]
E. The matrix is noninvertible.
Our calculated inverse matrix does not match any of the given choices exactly as written in options A, B, C, or D. Furthermore, option E which suggests the matrix is noninvertible contradicts our result showing that it is indeed invertible.
Hence, the correct understanding is:
None of the provided options accurately represent the inverse of the matrix.
