If A and B are square matrices of order 3 such that `|A|=5` and `AB=-5I`, then the value of `|B|` is

Options

A. `-5`
B. `-25`
C. 25
D. none of these

Answer

To solve the problem, we need to find the value of the determinant of matrix B given the conditions that |A| = 5 and AB = -5I, where I is the identity matrix of order 3.

### Step-by-Step Solution:

1. **Given Information**:
- |A| = 5
- AB = -5I (where I is the identity matrix of order 3)

2. **Taking Determinants**:
We start by taking the determinant of both sides of the equation AB = -5I.
\[ |AB| = |-5I| \]

3. **Using Properties of Determinants**:
The determinant of a product of matrices is equal to the product of their determinants:
\[ |AB| = |A| \cdot |B| \]
Therefore, we can write:
\[ |A| \cdot |B| = |-5I| \]

4. **Calculating the Determinant of -5I**:
The determinant of a scalar multiple of the identity matrix can be calculated using the formula:
\[ |kI| = k^n \quad \text{(where n is the order of the matrix)} \]
Here, k = -5 and n = 3 (since A and B are 3x3 matrices):
\[ |-5I| = (-5)^3 = -125 \]

5. **Substituting Values**:
Now we substitute the known values into the equation:
\[ |A| \cdot |B| = -125 \]
We know |A| = 5, so:
\[ 5 \cdot |B| = -125 \]

6. **Solving for |B|**:
To find |B|, we divide both sides by 5:
\[ |B| = \frac{-125}{5} = -25 \]

### Final Answer:
The value of |B| is -25.