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If we have an invertible matrix $M$ that is $4 \times 4$ and $\left| M \right| \neq 0$ (i.e. it is invertible), is it possible to decompose it into two matrices $4 \times 3$ and $3 \times 4$ $\left(A,B\right)$ respectively such that $AB = M$ , I just need an example if this possible, if not possible, can you argue?

Martin Sleziak
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islamfaisal
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  • http://math.stackexchange.com/questions/978/how-to-prove-and-interpret-operatornamerankab-leq-operatornamemin-ope, http://math.stackexchange.com/questions/48989/how-to-prove-textrankab-leq-min-textranka-textrankb – Martin Sleziak May 05 '15 at 10:24

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It is impossible. If $M$ is invertible it has rank 4, but $A$ and $B$ can both only have rank 3 or less.

sbares
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