Doubt in linear algebra Question (CSIR NET 2012)

Asked Dec 06 '22 at 15:54

Active Dec 06 '22 at 16:17

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Let $A$ and $B$ be the $n \times n $ real matrices such that $AB=0=BA$ and $A+B$ is invertible. Which of the following are always true?

1 $\mbox{rank}(A)=\mbox{rank}(B)$

2 $\mbox{rank}(A)+\mbox{rank}(B)=n$

3 $\mbox{nullity}(A)+\mbox{nullity}(B)=n$

4 $A-B$ is invertible.

option 1 is wrong if we take $A=0$ and $B=I_n$ but for rest options i stucked how to think ?

edited Dec 06 '22 at 16:17

asked Dec 06 '22 at 15:54

2

Your reasoning is incorrect. For one, note that $AB$ can be zero even when $A$ and $B$ are non-zero. For example, $$ A = \pmatrix{1&0\0&0}, \quad B = \pmatrix{0&0\0&1} $$ – Ben Grossmann Dec 06 '22 at 15:59
Also, it's not clear why you believe that $A + B$ being invertible implies that $A - B$ is invertible – Ben Grossmann Dec 06 '22 at 15:59
@BenGrossmann - Your counterexample was the one that came immediately to mind. Do people try to look at simple examples anymore? – Chris Leary Dec 06 '22 at 16:02
@BenGrossmann ohh yeah this was my mistake. further and i have edited my questions, thanks for comment. – Mathematics learner Dec 06 '22 at 16:07
@BenGrossmann but sir how to approach these kind of questions my reasoning is completely wrong any hint is appreciable – Mathematics learner Dec 06 '22 at 16:09

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