Question

If \( A = \begin{pmatrix} 0 & 1 & -2 -1 & 0 & 3 2 & -3 & 0 \end{pmatrix} \), then \( A^{-1} \)

• A. equal to \( -\frac{1}{12}(\text{adj }A) \)
• B. equal to \( -12 \)
• C. equal to \( \frac{1}{12}(\text{adj }A) \)
• D. doesn't exist

Hint:

Always check determinant first before finding inverse. If determinant is zero, inverse does not exist.

Answer 1

The Correct Option is D

Step 1: Recall the condition for existence of inverse.
A matrix \( A \) is invertible if and only if its determinant is non-zero:
\[ |A| \neq 0. \]

Step 2: Write the determinant of the matrix.
\[ |A| = \begin{vmatrix} 0 & 1 & -2 -1 & 0 & 3 2 & -3 & 0 \end{vmatrix}. \]

Step 3: Expand the determinant.
Using first row expansion:
\[ |A| = 0\begin{vmatrix} 0 & 3 -3 & 0 \end{vmatrix} -1\begin{vmatrix} -1 & 3 2 & 0 \end{vmatrix} -2\begin{vmatrix} -1 & 0 2 & -3 \end{vmatrix}. \]

Step 4: Compute minors.
\[ = -1((-1)(0) - (3)(2)) -2((-1)(-3) - 0). \]
\[ = -1(0 - 6) -2(3 - 0). \]
\[ = -1(-6) - 6. \]
\[ = 6 - 6 = 0. \]

Step 5: Interpret the result.
Since:
\[ |A| = 0, \]
the matrix is singular.

Step 6: Conclude about inverse.
A singular matrix does not have an inverse.

Step 7: Final conclusion.
Thus, \( A^{-1} \) does not exist.
Final Answer:
\[ \boxed{\text{doesn't exist}}. \]