Question

If \( B \) is a non-singular \(4\times4\) matrix and \( A \) is its adjoint such that \( |A|=125 \), then \( |B| \) is:

• A. \( 5 \)
• B. \( 25 \)
• C. \( 125 \)
• D. \( 625 \)

Hint:

For a square matrix \(A\) of order \(n\), always remember the important identity: \[ |\operatorname{adj}(A)|=|A|^{n-1} \] Special cases: \[ 2\times2 \Rightarrow |\operatorname{adj}(A)|=|A| \] \[ 3\times3 \Rightarrow |\operatorname{adj}(A)|=|A|^2 \] \[ 4\times4 \Rightarrow |\operatorname{adj}(A)|=|A|^3 \] This formula is extremely important in objective matrix questions.

Answer 1

The Correct Option is A

Concept: For a square matrix \(B\) of order \(n\), an important property relating the determinant of a matrix and the determinant of its adjoint is: \[ |\operatorname{adj}(B)|=|B|^{\,n-1} \] where:
• \( \operatorname{adj}(B) \) denotes the adjoint (adjugate) of matrix \(B\),
• \(n\) is the order of the matrix. This is one of the most important determinant properties used in matrix algebra. Since the matrix is non-singular: \[ |B|\neq 0 \] Therefore all determinant properties are valid.

Step 1: Identifying the order of the matrix We are given that \(B\) is a: \[ 4\times4 \] matrix. Hence the order of the matrix is: \[ n=4 \] Also \(A\) is the adjoint of \(B\). Thus: \[ A=\operatorname{adj}(B) \] We are given: \[ |A|=125 \]

Step 2: Using the determinant property of adjoint matrix The standard formula is: \[ |\operatorname{adj}(B)|=|B|^{n-1} \] Substituting \(n=4\): \[ |\operatorname{adj}(B)|=|B|^{4-1} \] \[ |\operatorname{adj}(B)|=|B|^3 \] But: \[ A=\operatorname{adj}(B) \] Therefore: \[ |A|=|B|^3 \] Since \( |A|=125 \), we get: \[ 125=|B|^3 \]

Step 3: Finding \( |B| \) Taking cube root on both sides: \[ |B|=\sqrt[3]{125} \] We know: \[ 125=5^3 \] Therefore: \[ |B|=5 \] Final Answer: \[ \boxed{|B|=5} \] Hence, the correct option is: \[ \boxed{(A)} \]