Question

If \(B\) is a non-singular \(4 \times 4\) matrix and \(A\) is its adjoint such that \(|A| = 125\), then \(|B|\) is:

• A. 5
• B. 25
• C. 125
• D.

Hint:

For an \(n \times n\) matrix \(B\), \(|\text{adj} B| = |B|^{n-1}\).

Answer 1

The Correct Option is A

Recall the relation between a square matrix \(B\), its adjoint \(A\), and their determinants: \[ |A| = |\text{adj} B| = |B|^{n-1} \] where \(n\) is the order of the matrix, here \(n=4\). Given, \[ |A| = 125, \quad n=4 \] So, \[ 125 = |B|^{4-1} = |B|^3 \] Take cube root on both sides: \[ |B| = \sqrt[3]{125} = 5 \]