Question

Let A be an invertible matrix then which of the following is/are true

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

Answer 1

Answer: (A), (B), (C)

The correct options are
A $(A^2{)}^{-1}=(A^{-1}{)}^2$
B $|A^{-1}|=|A{|}^{-1}$
C $(A^T{)}^{-1}=(A^{-1}{)}^T$

Option A

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

$det\left(A\right)\left(det\left(B\right)\right)$

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

$det\left(I\right)=1$

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

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

Option B:

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

$\Rightarrow A^2$ is also invertible

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

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

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

Option C:

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

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

Also,

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

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

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

Option A,B,C are correct