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: Comprehend the Problem:
The problem requires calculating the determinant of the adjugate matrix. A direct relationship exists between a matrix's determinant, its order, and the determinant of its adjugate.
Step 2: Identify the Governing Formula:
For any non-singular square matrix A with order n, the determinant of its adjugate is determined by:
\[ |\text{adj}(A)| = |A|^{n-1} \]Step 3: Apply the Formula with Given Data:
We are provided with:
Matrix A is non-singular.
Matrix A has an order \(n = 3\).
The determinant of matrix A is \(|A| = 15\).
Applying the formula from Step 2:
\[ |\text{adj}(A)| = |A|^{n-1} \]Substitute the known values:
\[ |\text{adj}(A)| = (15)^{3-1} \]\[ |\text{adj}(A)| = (15)^2 \]Step 4: Compute the Final Result:
Perform the calculation:
\[ (15)^2 = 225 \]Consequently, \(|\text{adj } A| = 225\).