Question

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

• A. $A^{-1}={\left|A\right|}^{-1}$
• B. ${\left(A^2\right)}^{-1}={\left(A^{-1}\right)}^2$
• C. ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$
• D. None of these

Answer 1

The correct option is A $A^{-1}={\left|A\right|}^{-1}$

1.

$A{A}^{-1}=I$

$\left|A{A}^{-1}\right|=1$

$\Rightarrow \left|A\right|\left|A^{-1}\right|=1$

$\Rightarrow \left|A^{-1}\right|={\left|A\right|}^{-1}$

Option A is false

2.

Now, $A{A}^{-1}=I$

${\left(A{A}^{-1}\right)}^2=I^2\Rightarrow A^2{\left(A^{-1}\right)}^2=I$

$\Rightarrow {\left(A^{-1}\right)}^2=\frac{I}{A^2}$

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

Thus, option B is True

3.

$I=A{A}^{-1}$

$\Rightarrow {\left(A{A}^{-1}\right)}^T=I^T\Rightarrow {\left(A^{-1}\right)}^T{A}^T=I$

$\Rightarrow {\left(A^{-1}\right)}^T={\left(A^T\right)}^{-1}$

Thus, option C is true.