What is the Adjoint of inverse of a matrix?

Question

I recently read that when we take the Adjoint of the inverse of a matrix we get the matrix itself i.e. $\operatorname{adj}(A^{-1})=A$ I am unable to prove the result. Help!

Answer 1

It is not true, I'm afraid.

If the matrix $A$ is invertible, then $$ A^{-1}=\frac{1}{\det A}\operatorname{Adj}(A) $$ and, in general, the relation $$ (\det A)I = \operatorname{Adj}(A)A $$ holds for all matrices. In particular, for $A^{-1}$ in place of $A$, $$ (\det(A^{-1}))I = \operatorname{Adj}(A^{-1})A^{-1} $$ so that $$ \operatorname{Adj}(A^{-1})=(\det A)^{-1}A $$

Thus we have $\operatorname{Adj}(A^{-1})=A$ if and only if $\det A=1$.

For instance, if $$ A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ then $$ A^{-1}= \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix} $$ so that $$ \operatorname{Adj}(A^{-1})= \begin{bmatrix} -1/2 & -1 \\ -3/2 & -2 \end{bmatrix}= \frac{1}{-2}\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$