Question

If A and B are invertible matrices, which one of the following statement is/are correct

• A. $Adj\left(A\right)=|A|A^{-1}$
• B. $det\left(A^{-1}\right)=|det\left(A\right){|}^{-1}$
• C. $(A+B{)}^{-1}=B^{-1}+A^{-1}$
• D. $(AB{)}^{-1}=B^{-1}{A}^{-1}$

Answer 1

Answer: (A), (B), (D)

$(AB{)}^{-1}=B^{-1}{A}^{-1}$
$Adj\left(A\right)=|A|A^{-1}$
$det\left(A^{-1}\right)=|det\left(A\right){|}^{-1}$

Option A

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

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

Option A is true

Option B

$det\left(AB\right)=\left(detA\right)\left(detB\right)$

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

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

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

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

Option B is true

Option C

${(A+B)}^{-1}=A^{-1}+B^{-1}$

Option C is true

Option D

${\left(AB\right)}^{-1}=?$

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

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

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

Option D is true