Question

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$, then the value of the determinant of $A^{-1}$ is

• A. $-6$
• B. $-\frac{1}{6}$
• C. $\frac{1}{36}$
• D. $36$

Hint:

Never spend time calculating the adjoint or inverse matrix if the question only asks for the determinant of an inverse or power of a matrix. Always use determinant properties like $|A^{-1}| = |A|^{-1}$ or $|A^n| = |A|^n$ to save valuable time.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Given a 3×3 matrix A, find the determinant of its inverse |A⁻¹|.

Step 2: Key Formula or Approach:
A fundamental determinant property states |A⁻¹| = 1/|A|. So we only need to compute the determinant of the original matrix A.

Step 3: Detailed Explanation:
Expanding |A| along the first row: |A| = 1·|2 3; 2 1| - 0 + 1·|0 2; 1 2| = 1·(2·1 - 3·2) + 1·(0·2 - 2·1) = (2 - 6) + (0 - 2) = -4 - 2 = -6. Therefore, |A⁻¹| = 1/(-6) = -1/6.

Step 4: Final Answer:
The determinant is -1/6, option (B).