Question

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^–1) = (det A)^k.

Answer 1

As we know that
 \[A^{- 1} = \frac{Adj A}{\left| A \right|}\]
\[ \therefore \left| A^{- 1} \right| = \frac{\left| Adj A \right|}{\left| A \right|}\]
\[ = \frac{\left| A \right|^{3 - 1}}{\left| A \right|} \left[ \because\text{ If A is a non singular matrix of order n, then }\left| adj\left( A \right) \right| = \left| A \right|^{n - 1} \right]\]
\[= \frac{\left| A \right|^2}{\left| A \right|}\]
\[ = \left| A \right|\]
\[\text{ As we are given that }\left| A^{- 1} \right| = \left| A \right|^k\]
\[\therefore k = 1\]