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Proof: rref[A|AB]=[In|B] regarding determinants

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The exercise wants us to show the following equality, by thinking in terms of block matrices.

We`re supposed to used the fact that [B; In] is in the kernel of [In; M]

rref[A AB] = [I M] and we must show the equality M=B.

I am pretty clueless on how to approach this. A link to the proof or some (solid) hints would be appreciated.