I have read that the product of two nonsingular matrices is nonsingular. So if $A$ and $B$ are nonsingular, then so is $AB$.
But what if we have $AC$ is nonsingular. Is there a rule in factorisation such that if we have a nonsingular matrix $AC$, then both $A$ and $C$ have to be nonsingular?
From my understanding you can have an $n \times n$ nonsingular matrix $A$ and a $n \times n$ nonsingular matrix $D$. So the multiplication would work, but because we have a nonsingular $\times$ singular, do we know what the result would be in terms of invertibility?
All I know is that a nonsingular $\times$ nonsingular $=$ nonsingular