Question

If A is an invertible matrix, then which of the following is not true ?

Options

• \[\left( A^2 \right)^{- 1} = \left( A^{- 1} \right)^2\]

• \[\left| A^{- 1} \right| = \left| A \right|^{- 1}\]

• \[\left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T\]

• \[\left| A \right| \neq 0\]

Answer 1

\[\left( A^2 \right)^{- 1} = \left( A^{- 1} \right)^2\]

We have, \[\left| A^{- 1} \right| = \left| A \right|^{- 1}\], \[\left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T\] and \[\left| A \right| \neq 0\] all are the properties of the inverse of a matrix A.