Question

If \(B\) is a non-singular matrix and \(A\) is a square matrix, then \(\det(B^{-1}AB)\) is equal to

• A. \(\det(A^{-1})\)
• B. \(\det(B^{-1})\)
• C. \(\det(A)\)
• D. \(\det(B)\)

Hint:

Similar matrices \(A\) and \(B^{-1}AB\) have the same determinant. Because \(\det(B^{-1}AB)=\det(B^{-1})\det(A)\det(B)=\det(A)\).

Answer 1

The Correct Option is C

Step 1: Use determinant property.
\[ \det(XYZ)=\det(X)\det(Y)\det(Z) \] Step 2: Apply to \(\det(B^{-1}AB)\).
\[ \det(B^{-1}AB)=\det(B^{-1})\det(A)\det(B) \] Step 3: Use \(\det(B^{-1})=\dfrac{1{\det(B)}\).}
\[ \det(B^{-1})\det(B)=1 \] Step 4: Final simplification.
\[ \det(B^{-1}AB)=\det(A) \] Final Answer: \[ \boxed{\det(A)} \]