Question

Match the properties of adjoint of a matrix :-

• A. ${\left(AdjA\right)}^T$
• B. $AdjB.AdjA$
• C. ${\left(A\right)}^{{}^{N-1}}$ ,N is order of $\left(A\right)$

Answer 1

$\to \mathrm{adj}A=\left(A\right|A^{-1}\left(\mathrm{adj}A\right|=\left(\left(A\right|A^{-1}\right|={\left(A\right|}^N\frac{1}{\left(}={\left(A\right|}^N{\left(A\right|}^{-1}={\left(A\right|}^{N-1}\to \mathrm{Adj}\mathrm{AB}=\left(\mathrm{AB}\right|{\left(\mathrm{AB}\right)}^{-1}=\left(A\right|\left(B\right|B^{-1}{A}^{-1}=\left(B\right|B^{-1}\left(A\right|A^{-1}=\mathrm{Adj}B.\mathrm{Adj}A\to \mathrm{Adj}\left(A^T\right)={\left(A^T\right)}^{-1}\left(A^T\right|={\left(A^{-1}\right)}^T{\left(A\right|}^T\left(\because {\left(A^{-1}\right)}^T=A^T,\left(A^T\right|={\left(A\right|}^T\right){\left(\left(A\right|A^{-1}\right)}^T={\left(\mathrm{Adj}A\right)}^T$