Question

If A and B are square matrices of order 3 such that \(|A| = -1\), \(|B| = 3\) then \(|3AB|\) is:

• A. \(-9\)
• B. \(-81\)
• C. \(-27\)
• D. \(81\)

Hint:

For an \(n \times n\) matrix \(A\), \(|kA| = k^n |A|\), where \(k\) is a scalar.

Answer 1

The Correct Option is B

Given \(|A| = -1\) and \(|B| = 3\), both matrices are of order 3. Using properties of determinants: \[ |3AB| = |3I \cdot A \cdot B| = |3I| \cdot |A| \cdot |B| \] where \(3I\) is the scalar matrix \(3\) times the identity matrix of order 3. \[ |3I| = 3^3 = 27 \] Therefore, \[ |3AB| = 27 \times (-1) \times 3 = -81 \]