Question

If A is invertible, then 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. ${\left(A^{{}^{\prime }}\right)}^{-1}={\left(A^{-1}\right)}^{{}^{\prime }}$
• D. None of these

Answer 1

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

A is invertible

$\Rightarrow A^{-1}$ exists

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

But we cannot write that a matrix and its determinant are both equal

$\therefore$ option A is not true

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

This option is true from the property

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

Option C: ${\left(A^{-1}\right)}^1={\left(A^1\right)}^{-1}$

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

Similarly

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

From $1$ and $2$

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

$\Rightarrow A^1$ is multiplicative inverse of ${\left(A^{-1}\right)}^1$

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