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I would like you to help me with two questions I am stuck in. You can call these homework questions. It would be helpful if you can give me non-trivial hints instead of complete solution.

1) Let $A$ be a matrix such that $Rowspace(A)=Coulmnspace(A)$. Prove or disprove that A is symmetric.

2) Suppose $A$ and $B$ are two matrices such that $Columnspace(A)=Columnspace(B)$ and $BA=AB$. Prove or disprove that $A=B$

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Counterexample for both:

$$ A = \pmatrix{0&1&0\\0&0&1\\1&0&0}, \quad B = \pmatrix{1&0&0\\0&1&0\\0&0&1} $$

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  • $\begingroup$ Thanks! Well I was assuming these are true and hence trying to prove it, which of course I couldn't. But I was under the impression that if I could not, maybe someone else can prove they are true! But in general, while A and B turn out to be pretty commonplace matrices,what would you suggest me when I encounter such "Prove or disprove" questions? Should I first check the identity, null, permutation, Kronecker and 1 matrices before trying to prove? $\endgroup$ – Landon Carter Oct 31 '14 at 11:52
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    $\begingroup$ I started the first one by checking the statement "if $A$ has the same rows as it has columns, then $A$ is symmetric", either to . Really, for that first problem, any non-singular, non-symmetric, square matrix works just as well. $\endgroup$ – Omnomnomnom Oct 31 '14 at 11:57
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    $\begingroup$ I think the smarter bet is to just start with $2 \times 2$ matrices. For example, we could have had $$A = \pmatrix{1&1\\0&1}$$ for the first, and the same $A$ with $$ B = \pmatrix{1&0\\0&1} $$ for the second. For any pair of matrices, a good way to start is always "what if one is the identity?" $\endgroup$ – Omnomnomnom Oct 31 '14 at 11:59
  • $\begingroup$ I shall keep this in mind. Often I get bogged down by these kind of prove or disprove type of questions. $\endgroup$ – Landon Carter Oct 31 '14 at 12:03
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  1. Two vectors can be composed from the same basis, but not be equal.
  2. Did you mis-transcribe the question? Equality is symmetric for matrices. BA $=$ AB $\implies$ AB $=$ BA
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  • $\begingroup$ Indeed the question was incorrect in part 2. I have edited and written the correct question. So sorry for this inconvenience! $\endgroup$ – Landon Carter Oct 31 '14 at 11:21
  • $\begingroup$ Meanwhile, would you be kind enough to re-explain your hint to question 1? I really can't figure out your hint. Thanks for your time! $\endgroup$ – Landon Carter Oct 31 '14 at 11:25
  • $\begingroup$ Looks like omnomnomnom gave you the answer. $\endgroup$ – Zaaier Oct 31 '14 at 20:20

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