# An inverse of the adjoint

#### Yankel

##### Active member
Hello

I need some help proving the next thing, I can't seem to be able to work it out..

Let A be an nxn matrix.

Prove that:

$$(adj A)^{-1} = adj(A^{-1})$$

Thanks...

#### Deveno

##### Well-known member
MHB Math Scholar
$A = IA = A^*(A^*)^{-1}A$

so:

$A^* = (A^*(A^*)^{-1}A)^* = A^*((A^*)^{-1})^*A$

therefore:

$A^*A^{-1} = A^*((A^*)^{-1})^*$

and multiplying on the left by $(A^*)^{-1}$ we get:

$A^{-1} = ((A^*)^{-1})^*$

so

$(A^{-1})^* = ((A^*)^{-1})^{**} = (A^*)^{-1}$

#### Yankel

##### Active member
thanks, took me some time to understand your proof, but now I see it, nice one !