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## Main Question or Discussion Point

Say I have a mxn matrix A and a nxk matrix B. How do you prove that the matrix C = AB is full-rank, as well?

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Say I have a mxn matrix A and a nxk matrix B. How do you prove that the matrix C = AB is full-rank, as well?

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chiro

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For this problem, I'm wondering if you can show that the null-space for the final space is directly related to the null-spaces of the original two systems (like for example if they are additive or have some other relation).

If this is possible, then if you can show that the nullity is zero then you have shown it has full rank.

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What do you mean with "full rank" in the first place?

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By full-rank, I mean rank(M) = min{num_cols(M), num_rows(M)}.

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In that case, the statement in the OP is false. Try to find a counterexample.By full-rank, I mean rank(M) = min{num_cols(M), num_rows(M)}.

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mathwonk

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- #7

AlephZero

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That's true for square matrices, but the OP's matrices are not necessariily square (again, assuming it was stated correctly).

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