1. Inverses of matrices

How does the identity A(I+BA)=(I+AB)A connect the inverses of I+BA and I+AB? Prove that they are both invertible or both singular.

2. Originally Posted by alexmahone
How does the identity A(I+BA)=(I+AB)A connect the inverses of I+BA and I+AB? Prove that they are both invertible or both singular.
Here is an idea, if we know something about if $A$. If $A$ is invertible $(I+BA)=A^{-1}(I+AB)A$ then

This would say that $I+BA$ is similar to $I+AB$. Since similar Matrices have the same determinant They are both either singular or invertible.

3. Originally Posted by alexmahone
How does the identity A(I+BA)=(I+AB)A connect the inverses of I+BA and I+AB? Prove that they are both invertible or both singular.

In the general case, that is $A$ singular or not, suppose ( without loss of generality ) $I+BA$ singular and $I+AB$ invertible, then:

$(i)\quad \left |{I+BA}\right |=0 \Leftrightarrow \left |{BA-(-1)I}\right |=0 \Leftrightarrow -1\in\textrm{spec}(BA)$

$(ii)\quad \left |{I+AB}\right |\neq 0 \Leftrightarrow \left |{AB-(-1)I}\right |\neq 0 \Leftrightarrow -1 \notin \textrm{spec}(AB)$

$\textrm{spec}(BA)=\textrm{spec}(AB)$