Question

If $ A $ is a $ 2 \times 2 $ matrix and $ |A| = 4 $, then $ |A^{-1}| $ is:

• A. 16
• B. \( \frac{1}{4} \)
• C. 4
• D. 1

Hint:

The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix. This property holds for all square matrices.

Answer 1

The Correct Option is B

Given a \( 2 \times 2 \) matrix \( A \) with \( |A| = 4 \), we seek \( |A^{-1}| \).According to matrix theory, the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix:\[|A^{-1}| = \frac{1}{|A|}\]Substituting the provided value \( |A| = 4 \):\[|A^{-1}| = \frac{1}{4}\]Therefore, the determinant of \( A^{-1} \) is \( \frac{1}{4} \).