$AB$ is invertible but $A$ or $B$ is not invertible.
108781
$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?
42809
How can we show that $(I-A)$ is invertible?
18585
Showing $A-I$ is invertible, when $A$ is a skew-symmetric matrix
118981
Proving $\sqrt{ab} = \sqrt a\sqrt b$
36998
In $Ax=b$. If $A$ is not invertible there are no solutions or infinity. How to determine what the case is?
12927
$A^2+B^2=AB$ and $BA-AB$ is non-singular
45523
Can the product of two non invertible elements in a ring be invertible?
98963
Prove that neither $A$ nor $B$ is divisible by $5$
110944
Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$
83475
Prove Or Disprove: tr(AB)=tr(A)*tr(B)
89177
$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$
85032
$Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
75066
If the product of two square matrices is invertible, then both matrices are invertible
15416
A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from "Topics in Matrix Analysis" for matrices of algebraic numbers.
78736
Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
6096
If $A+B=AB$ then $AB=BA$
13012
Why can a = b imply 2 = 1?