Let A be an invertible matrix. Which of the following is not true?

A
${(A^{T})}^{- 1} = {(A^{- 1})}^{T}$
B
$A^{- 1} = | A |^{- 1}$
C
$A^{- 2} = | A |^{- 1}$
D
$|A^{- 1}| = | A |^{- 1}$

Solution:

We know that ${(A^{T})}^{- 1} = {(A^{- 1})}^{T}$

So, option (a) is true.

In option $(b), A^{- 1}$ is a matrix and $| A |^{- 1}$ is a number. So, it is not true.

Now,

$\begin{array}{l} A^{2} {(A^{- 1})}^{2} = (A A) (A^{- 1} A^{- 1}) \\ \Rightarrow A^{2} {(A^{- 1})}^{2} = A (A A^{- 1}) A^{- 1} = (A I) A^{- 1} = A A^{- 1} = I \\ ∴ {(A^{2})}^{- 1} = {(A^{- 1})}^{2} \end{array}$

So, option (c) is true.

Solution:

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