If B is a non-singular matrix and A is a square matrix, then the value of det `(B^(-1) AB)` is equal to :

Options

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

Answer

To find the value of \( \det(B^{-1}AB) \), we will use properties of determinants. Here’s the step-by-step solution:

### Step 1: Use the property of determinants
We know that the determinant of a product of matrices is equal to the product of their determinants. Therefore, we can write:
\[ \det(B^{-1}AB) = \det(B^{-1}) \cdot \det(A) \cdot \det(B) \]

### Step 2: Find the determinant of \( B^{-1} \)
The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix. Thus, we have:
\[ \det(B^{-1}) = \frac{1}{\det(B)} \]

### Step 3: Substitute back into the equation
Now we can substitute \( \det(B^{-1}) \) back into our equation from Step 1:
\[ \det(B^{-1}AB) = \left(\frac{1}{\det(B)}\right) \cdot \det(A) \cdot \det(B) \]

### Step 4: Simplify the expression
In the expression above, \( \det(B) \) in the numerator and denominator will cancel out:
\[ \det(B^{-1}AB) = \frac{\det(A) \cdot \det(B)}{\det(B)} = \det(A) \]

### Final Answer
Thus, the value of \( \det(B^{-1}AB) \) is:
\[ \det(B^{-1}AB) = \det(A) \]

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