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The Maximum Rank of a Matrix B Given AB=0 and A is a Full Rank Matrix
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[QUOTE="fresh_42, post: 6029879, member: 572553"] In this case you couldn't multiply the matrices in the first place. This means for the transformations, that we need to explain, what happens between the 8-dimensional image of ##B## and the 7-dimensional space, on which ##A## is defined. You can translate it into row and column actions, too. But as we don't have a certain example, we can assume that the matrices are already in a form which is nice: $$ A=\begin{bmatrix}I_3 & | & 0_4 \end{bmatrix}\, , \,B=\begin{bmatrix} B_7^{\,'}&| &0_{46}\end{bmatrix} $$ with the ##(3\times 3)## identity matrix ##I_3## and any ##(7 \times 7)## matrix ##B_7^{\,'}##. This wouldn't change the result but is easier to handle. Now calculate ##AB =0## and see what does this mean for ##B_7^{\,'}## and its maximal rank. Here are a couple of formulas which also might help occasionally: [URL]https://en.wikipedia.org/wiki/Rank_(linear_algebra)#Properties[/URL] In your example above I basically used [URL='https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem']https://en.wikipedia.org/wiki/Rank–nullity_theorem[/URL] [/QUOTE]
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The Maximum Rank of a Matrix B Given AB=0 and A is a Full Rank Matrix
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