Question

If \[ A = \begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix}, \] then \( A \cdot \text{(adj A)} \) is equal to:

• A. \( A \)
• B. \( |A| \)
• C. \( |A| \)
• D. None of these

Hint:

For any square matrix \( A \), the relation \( A \cdot \text{adj}(A) = |A| \cdot I \) is always true.

Answer 1

The Correct Option is B

Step 1: Understanding the formula.
For any square matrix \( A \), the relation \( A \cdot \text{adj}(A) = |A| \cdot I \) holds, where \( I \) is the identity matrix and \( |A| \) is the determinant of matrix \( A \). Step 2: Analyzing the options.
- (1) \( A \): This is incorrect because \( A \cdot \text{adj}(A) \) does not give \( A \).
- (2) \( |A| \): This is correct as per the formula \( A \cdot \text{adj}(A) = |A| \cdot I \).
- (3) \( |A| \): This is a repetition of option (2) and incorrect in this context.
- (4) None of these: This is incorrect as option (2) is correct.
Conclusion.
The correct answer is (2) \( |A| \), as the determinant of matrix \( A \) is the result when multiplying \( A \) by its adjugate matrix.