If B is a non singular matrix and A is a square matrix, the `det(B^(-1)AB)=`

Options

A. `det(A^(-1))`
B. `det(B^(-1))`
C. `det(A)`
D. `det(B)`

Answer

To find the determinant of the matrix expression \( \det(B^{-1}AB) \), where \( B \) is a non-singular matrix and \( A \) is a square matrix, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Non-Singularity**:
Since \( B \) is a non-singular matrix, it means that \( \det(B) \neq 0 \). This implies that the inverse \( B^{-1} \) exists.

**Hint**: Recall that a matrix is non-singular if its determinant is not zero.

2. **Using Determinant Properties**:
We can use the property of determinants that states:
\[ \det(AB) = \det(A) \cdot \det(B) \]
for any two square matrices \( A \) and \( B \).

**Hint**: Remember that the determinant of a product of matrices is the product of their determinants.

3. **Applying the Property**:
We want to find \( \det(B^{-1}AB) \). We can rewrite this using the property mentioned:
\[ \det(B^{-1}AB) = \det(B^{-1}) \cdot \det(A) \cdot \det(B) \]

**Hint**: Break down the expression into parts using the determinant property.

4. **Finding \( \det(B^{-1}) \)**:
We know from the properties of determinants that:
\[ \det(B^{-1}) = \frac{1}{\det(B)} \]

**Hint**: The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix.

5. **Substituting Back**:
Now substituting \( \det(B^{-1}) \) into our expression:
\[ \det(B^{-1}AB) = \left(\frac{1}{\det(B)}\right) \cdot \det(A) \cdot \det(B) \]

**Hint**: Simplify the expression by substituting the known value of \( \det(B^{-1}) \).

6. **Simplifying the Expression**:
The \( \det(B) \) in the numerator and denominator cancels out:
\[ \det(B^{-1}AB) = \det(A) \]

**Hint**: Look for opportunities to cancel terms to simplify your calculations.

### Final Result:
Thus, we conclude that:
\[ \det(B^{-1}AB) = \det(A) \]

### Summary:
The determinant of the matrix \( B^{-1}AB \) is equal to the determinant of \( A \), regardless of the non-singular matrix \( B \).