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Let $A$, $B$ be $n*n$ matrices such that $BA +B^2=I-BA^2$ where $I$ is the $n*n$ matrix.Which of the following is always true?

1) $A$ is non singular

2) $B$ is non singular

3) $A+B$ is non singular

4) $AB$ is non singular

We've $BA +B^2=I-BA^2$. Hence $B(A +B+A^2)=I$.

So can we say that $B$ is non singular?

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    $\begingroup$ Yes, $B$ is nonsingular. Are you supposed to choose only one answer? $\endgroup$ – Valentin May 7 '15 at 13:16
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Yes $B$ is non singular and for the others take $B:=-I$ and $A:=0$ then it satisfies the equation so 1) and 4) cannot be true in general. If you take $B:=I$ and $A:=-I$ then $BA+B^2=0$ and $I-BA^2=0$ so the equation is true but $A+B=0$ so 3) cannot be true in general.

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