Find the inverse, if it exists, for the matrix.
-$$
\left[\begin{array}{rrr}
2 & -1 & 0 \
3 & -2 & 0 \
-2 & 3 & 1
\end{array}\right]
$$
A)$ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & 2 & 0 \ -5 & 4 & 1\end{array}\right] $
B)$ \left[\begin{array}{rrr}1 & -1 & 0 \ 3 & -2 & 1 \ -5 & 4 & -1\end{array}\right] $
C) $ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & -2 & 0 \ -5 & 4 & -1\end{array}\right] $
D)$ \left[\begin{array}{rrr}2 & -1 & 0 \ 3 & -2 & 0 \ -5 & 4 & 1\end{array}\right] $

Question

Quizplus · Accepted Answer

To find the inverse of a matrix, one common method is to apply the Gauss-Jordan elimination on the augmented matrix [A|I], where A is the given matrix and I is the identity matrix. However, for this specific question, we can observe that the third column of the given matrix is already in the form of an identity matrix's third column. This suggests that the transformation affecting the first two columns does not impact the third column, implying that the inverse matrix will maintain the identity matrix's third column. The correct inverse matrix must therefore have a third column of [0, 0, 1]. Among the options, only option D has this property and also correctly adjusts the first two columns to reflect the inverse transformations needed to revert the effects of the original matrix on any vector.

Find the Inverse, If It Exists, for the Matrix [2−103−20−231]\left[\begin{array}{rrr}2 & -1 & 0 \\3 & -2 & 0 \\-2 & 3 & 1\end{array}\right]​23−2​−1−23​001​​

Find the Inverse, If It Exists, for the Matrix $\left[\begin{array}{rrr}2 & -1 & 0 \\3 & -2 & 0 \\-2 & 3 & 1\end{array}\right]$