Question

If, A is a square matrix of order 3 and |A| = -2 then. \(|-2 \ A^{-1}|\) is:

• A. -4
• B. 4
• C. 8
• D. 2

Answer 1

The Correct Option is B

To find \(|-2 \cdot A^{-1}|\) for a square matrix \(A\) of order 3 with \(|A| = -2\), we use the properties of determinants:

• The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix: \(|A^{-1}| = \frac{1}{|A|}\).
• Therefore, \(|A^{-1}| = \frac{1}{-2} = -\frac{1}{2}\).
• For any scalar \( k \) and matrix \( M \), the determinant is \(|kM| = k^n |M|\) where \(n\) is the order of the matrix.
• So, \(|-2 \cdot A^{-1}| = (-2)^3 \cdot |A^{-1}| = -8 \cdot \left(-\frac{1}{2}\right) = 4\).

Thus, the value of \(|-2 \cdot A^{-1}|\) is 4.