Question

If A is a square matrix of order 3 and |A| = 5, then |A adj.A| is

• A. 5
• B. 125
• C. 25
• D. 625

Answer 1

The Correct Option is B

If A is a square matrix of order 3 and |A| = 5, then we need to find |A adj(A)|.

We know that \(A \cdot adj(A) = |A|I\), where I is the identity matrix.

So, |A adj(A)| = ||A|I|

Since A is of order 3, \(|A| = 5\), and I is the 3x3 identity matrix:

\(|A adj(A)| = |5I| = | \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}|\)

\(|5I| = 5^3 |I| = 5^3 (1) = 125\)

Therefore, |A adj(A)| = 125.

Thus, the correct option is (B) 125.

Answer 2

Step 1: Property of Determinant

For a square matrix $ A $ of order $ n $, the following property holds:

$$ |A \, \text{adj}(A)| = |A| \times |\text{adj}(A)| $$

For an $ n \times n $ matrix, the determinant of the adjugate matrix $ \text{adj}(A) $ is related to $ |A| $ by the formula:

$$ |\text{adj}(A)| = |A|^{n-1} $$

For a matrix of order 3, $ n = 3 $, so:

$$ |\text{adj}(A)| = |A|^2 $$

Step 2: Calculate $ |A \, \text{adj}(A)| $

Given $ |A| = 5 $, we find:

$$ |\text{adj}(A)| = 5^2 = 25 $$

Now, using the property:

$$ |A \, \text{adj}(A)| = |A| \times |\text{adj}(A)| = 5 \times 25 = 125 $$

Conclusion: The value of $ |A \, \text{adj}(A)| $ is 125.