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Question
If A is an invertible matrix, then which of the following is not true ?
Options
\[\left( A^2 \right)^{- 1} = \left( A^{- 1} \right)^2\]
\[\left| A^{- 1} \right| = \left| A \right|^{- 1}\]
\[\left( A^T \right)^{- 1} = \left( A^{- 1} \right)^T\]
\[\left| A \right| \neq 0\]
Solution
\[\left( A^2 \right)^{- 1} = \left( A^{- 1} \right)^2\]
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.
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