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

Step 1: Understanding the Concept:
This question tests a specific property of determinants relating a matrix to its adjoint (also known as the adjugate).
For any square matrix \(M\) of order \(n\), the determinant of its adjoint is given by the determinant of the matrix raised to the power of \((n-1)\).
A non-singular matrix means its determinant is not zero, which allows us to use this power rule reliably.
The order of the matrix (\(n\)) is a key variable in this formula.
Step 2: Key Formula or Approach:
The formula required is:
\[ |\text{adj}(M)| = |M|^{n-1} \]
In this problem:
- \(M = B\)
- \(n = 4\) (since it is a \(4 \times 4\) matrix)
- \(\text{adj}(B) = A\)
Step 3: Detailed Explanation:
Let's substitute the given information into our formula:
\[ |\text{adj}(B)| = |B|^{n-1} \]
Substituting \(A\) for \(\text{adj}(B)\):
\[ |A| = |B|^{4-1} \]
\[ |A| = |B|^3 \]
We are given that \(|A| = 125\). Thus:
\[ 125 = |B|^3 \]
To find \(|B|\), we take the cube root of 125:
\[ |B| = \sqrt[3]{125} \]
We know that:
\[ 5 \times 5 \times 5 = 125 \]
Therefore:
\[ |B| = 5 \]
Step 4: Final Answer:
The determinant of matrix B is 5.
This matches Option (A).