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$A$ is defined as an $m\times m$ matrix which is not invertible. How can i show that there is an $m\times m$ matrix $B$ where $AB = 0$ but $B$ is not equal to $0$?

For the solution of this question I think giving an example is not enough because it is too easy to solve this by giving an example, so how can I show that $B$ is not the $0$ matrix?

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    Hint: $AB=0$ if and only if every column of $B$ is in the null space of $A$.2012-11-23

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