Question

If B is a non-singular matrix and A is a square matrix, then det (B⁻¹ AB) is equal to ___________ .

Options

• Det (A⁻¹)

• Det (B⁻¹)

• Det (A)

• Det (B)

Answer 1

Det (A)

B is non - singular.

\[\text{ This implies that }\left| B \right| \neq 0,\text{ that B is invertible and that }B^{- 1}\text{ exists }. \]

Here, B is invertible .

\[ \therefore \left| B^{- 1} \right| = \left| B \right|^{- 1} = \frac{1}{\left| B \right|}\]

\[ \Rightarrow \left| B^{- 1} AB \right| = \left| B^{- 1} \right|\left| AB \right|\]
\[ \Rightarrow \left| B^{- 1} AB \right| = \left| B \right|^{- 1} \left| A \right|\left| B \right| \]
\[ \Rightarrow \left| B^{- 1} AB \right| = \frac{1}{\left| B \right|}\left| A \right|\left| B \right| \]
\[ \Rightarrow \left| B^{- 1} AB \right| = \left| A \right|\]