Do imaginary inverses of non-invertible matrices exist?

Question

There isn't a real solution to $x^2 = -1$, but a complex solution $x = i$ exists. Similarly, does there exist a complex inverse of non-invertible matrices?

Answer 1

Suppose a real matrix $A$ has a complex inverse $U$: we have a real $n\times n$ matrix $A$ and a complex $n\times n$ matrix $U$, such that $AU=I$. Decompose $U$ into its real and imaginary parts, $U=V+iW$, where $V$ and $W$ are real. Then we get $A(V+iW)=I$, so $AV+iAW=I$, where $A, V, W$ are real. So we can equate real and imaginary parts:

$AV=I$
$AW=0$

So $V$ is the inverse of $A$. In other words, $A$ is invertible!