Question

If \(A\) is a square matrix of order \(3 \times 3\), and \(|\text{adj } A| = 25\), then the value of \(|2A|\) is

• A. 20
• B. \(\pm 20\)
• C. 40
• D. \(\pm 40\)

Hint:

Use \(|\text{adj } A| = |A|^2\) for order 3, then \(|2A| = 8|A|\).

Answer 1

The Correct Option is D

Step 1: Recall the formulas:
For an \(n \times n\) matrix, \(|\text{adj } A| = |A|^{n-1}\) and \(|kA| = k^n |A|\). Here \(n = 3\).

Step 2: Find \(|A|\):
\(|\text{adj } A| = |A|^2 = 25\). So \(|A| = 5\) or \(|A| = -5\). Both values are possible because a square is positive either way.

Step 3: Find \(|2A|\):
\[ |2A| = 2^3 |A| = 8|A| \] For \(|A| = 5\) this is 40. For \(|A| = -5\) this is \(-40\). So \(|2A| = \pm 40\).

Step 4: Check the other options:
Options 1 and 2 use 20, which would come from \(4|A|\) (the wrong power of 2). Option 3 gives only the positive case and ignores \(|A| = -5\).

Final Answer:
\(|2A| = \pm 40\), option 4. \[ \boxed{\pm 40} \]