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

The relationship between a square matrix \(B\), its adjoint \(A\), and their determinants is expressed as:\[|A| = |\text{adj} B| = |B|^{n-1}\]Here, \(n\) denotes the order of the matrix, with \(n=4\).We are given:\[|A| = 125, \quad n=4\]Substituting these values into the equation:\[125 = |B|^{4-1} = |B|^3\]To find \(|B|\), we take the cube root of both sides:\[|B| = \sqrt[3]{125} = 5\]