Suppose A is a matrix of order 3 and `B=|A|A^(-1)`. If `|A|=5`, then `|B|` is equal to

Options

A. `1`
B. `-5`
C. `-1`
D. `25`

Explanation

To solve the problem, we need to find the determinant of matrix \( B \), where \( B = |A| A^{-1} \) and \( |A| = 5 \). Here are the steps to find \( |B| \):

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \( A \) is a matrix of order 3, and the determinant of \( A \) is given as \( |A| = 5 \).

2. **Expressing Matrix \( B \)**:
The matrix \( B \) is defined as:
\[ B = |A| A^{-1} \]
Substituting the value of \( |A| \):
\[ B = 5 A^{-1} \]

3. **Finding the Determinant of \( B \)**:
We use the property of determinants that states:
\[ |cA| = c^n |A| \]
where \( c \) is a scalar and \( n \) is the order of the matrix \( A \). In our case, \( c = 5 \) and \( n = 3 \):
\[ |B| = |5 A^{-1}| = 5^3 |A^{-1}| \]

4. **Finding the Determinant of \( A^{-1} \)**:
The determinant of the inverse of a matrix is given by:
\[ |A^{-1}| = \frac{1}{|A|} \]
Therefore:
\[ |A^{-1}| = \frac{1}{5} \]

5. **Substituting Back to Find \( |B| \)**:
Now substituting \( |A^{-1}| \) back into the equation for \( |B| \):
\[ |B| = 5^3 \cdot \frac{1}{5} = 125 \cdot \frac{1}{5} = 25 \]

6. **Final Result**:
Thus, the determinant of matrix \( B \) is:
\[ |B| = 25 \]

### Summary:
The determinant \( |B| \) is equal to 25.