Question

If \( A \) is a non-singular square matrix of order \( 3 \times 3 \) such that its determinant is \( |A| = 5 \), find the absolute value of the determinant of its adjoint matrix, represented as \( |\text{adj}(A)| \).

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

Hint:

Keep this related identity handy for multi-step matrix problems: the determinant of the adjoint of an adjoint matrix scales even higher, following the rule \( |\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2} \).

Answer 1

The Correct Option is C

Concept: For any non-singular square matrix \( A \) of order \( n \times n \), the determinant of its adjoint matrix is directly related to the determinant of the original matrix through the fundamental algebraic identity: \[ |\text{adj}(A)| = |A|^{n-1} \] This property allows us to calculate the value without needing to compute the individual elements of the adjoint matrix.

Step 1: Identify the matrix order and determinant from the problem values.
The problem provides the following parameters:

• Matrix order (\( n \)) = 3
• Determinant value (\( |A| \)) = 5

Step 2: Substitute parameters into the adjoint determinant identity.
Plug these values directly into the scaling formula: \[ |\text{adj}(A)| = 5^{3-1} = 5^2 \]

Step 3: Simplify the exponential expression.
Evaluating the exponent yields the final scalar determinant value: \[ |\text{adj}(A)| = 25 \]