Question

Let A be a non-singular matrix of order 3 and \(|A| = 15\), then \(|\text{adj } A|\) is equal to

• A. 15
• B. 45
• C. 225
• D. 150

Hint:

This is a direct formula-based question. Memorize the properties of determinants and adjugate matrices, such as \(|\text{adj}(A)| = |A|^{n-1}\) and \(A(\text{adj } A) = (\text{adj } A)A = |A|I\). These are frequently tested.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the determinant of the adjugate (or adjoint) of a matrix. There is a direct relationship between the determinant of a matrix, its order, and the determinant of its adjugate.
Step 2: Key Formula or Approach:
For any non-singular square matrix A of order n, the determinant of its adjugate is given by the formula:
\[ |\text{adj}(A)| = |A|^{n-1} \] Step 3: Detailed Explanation:
We are given the following information:
The matrix A is non-singular.
The order of the matrix A is \(n = 3\).
The determinant of the matrix A is \(|A| = 15\).
Using the formula from Step 2:
\[ |\text{adj}(A)| = |A|^{n-1} \] Substitute the given values into the formula:
\[ |\text{adj}(A)| = (15)^{3-1} \] \[ |\text{adj}(A)| = (15)^2 \] Step 4: Final Answer:
Calculate the final value:
\[ (15)^2 = 225 \] Therefore, \(|\text{adj } A|\) is equal to 225.