If A is a square matrix of order `2times2 and |A|=5,` then `|A(adjA)|`=

Options

A. `5`
B. `20`
C. `25`
D. `30`

Answer

To solve the problem, we need to find the value of \(|A(\text{adj} A)|\) given that \(|A| = 5\) and \(A\) is a \(2 \times 2\) matrix.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that for any square matrix \(A\), the product of \(A\) and its adjugate (adjoint) is given by:
\[ A \cdot \text{adj} A = |A| \cdot I_n \]
where \(I_n\) is the identity matrix of order \(n\).

2. **Substituting the Determinant**:
Since we are given that \(|A| = 5\) and \(A\) is a \(2 \times 2\) matrix (so \(n = 2\)), we can substitute this into the equation:
\[ A \cdot \text{adj} A = 5 \cdot I_2 \]

3. **Taking the Determinant of Both Sides**:
Now, we take the determinant of both sides:
\[ |A \cdot \text{adj} A| = |5 \cdot I_2| \]

4. **Using the Property of Determinants**:
The property of determinants states that:
\[ |A \cdot B| = |A| \cdot |B| \]
Therefore, we can write:
\[ |A \cdot \text{adj} A| = |A| \cdot |\text{adj} A| \]

5. **Finding the Determinant of the Right Side**:
The determinant of a scalar multiple of a matrix can be computed as:
\[ |k \cdot I_n| = k^n \cdot |I_n| \]
Here, \(k = 5\) and \(n = 2\):
\[ |5 \cdot I_2| = 5^2 \cdot |I_2| = 25 \cdot 1 = 25 \]

6. **Equating the Determinants**:
Now we equate both sides:
\[ |A| \cdot |\text{adj} A| = 25 \]

7. **Finding the Determinant of the Adjugate**:
For a \(2 \times 2\) matrix, the determinant of the adjugate is related to the determinant of the original matrix:
\[ |\text{adj} A| = |A|^{n-1} \]
Here, \(n = 2\), so:
\[ |\text{adj} A| = |A|^{2-1} = |A|^1 = |A| = 5 \]

8. **Final Calculation**:
Now substituting \(|A|\) back into our equation:
\[ |A| \cdot |\text{adj} A| = 5 \cdot 5 = 25 \]

Thus, we conclude that:
\[ |A(\text{adj} A)| = 25 \]

### Final Answer:
\[ |A(\text{adj} A)| = 25 \]