Question

Let \( A \) be a square matrix of order 3 such that \( |A| = 5 \). Find the value of the determinant of its adjoint matrix, \( |\text{adj}(A)| \).

• A. \( 25 \)
• B. \( 5 \)
• C. \( 125 \)
• D. \( 15 \)

Hint:

The exponent for the adjoint property is always one less than the matrix order (\( n-1 \)). For a \( 3 \times 3 \) matrix, simply square the original determinant value to find the answer immediately.

Answer 1

The Correct Option is A

Concept: The determinant of an adjoint matrix is directly linked to the determinant of the original matrix through the standard exponent property: \[ |\text{adj}(A)| = |A|^{n-1} \] where \( n \) represents the spatial order of the square matrix.

Step 1: Identify the order and original determinant value. From the problem description:
• Matrix order, \( n = 3 \)
• Original determinant value, \( |A| = 5 \)

Step 2: Substitute the values into the adjoint property formula. Plugging our isolated metrics into the exponent template: \[ |\text{adj}(A)| = 5^{3-1} \] Simplify the exponent calculation: \[ |\text{adj}(A)| = 5^2 = 25 \] Thus, the determinant value of the adjoint matrix is exactly 25.