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Rank of product of matrices

Let A, B be m \times n and n \times k matrices respectively with entries over some field. Prove that

    \[{\rm rank} (AB) \geq {\rm rank} (A) + {\rm rank}(B) -n\]


We begin by stating a lemma:

Lemma: It holds that

    \[{\rm nul} (T_1 T_2) \leq {\rm nul} (T_1) + {\rm nul} (T_2)\]

where T_1, \; T_2 are the corresponding linear transformations.

The proof of the lemma is based on the rank – nullity theorem.

Based upon the above lemma we have that

    \begin{align*} {\rm rank} \left ( T_1 T_2 \right ) + n &= k - {\rm nul} \left ( T_1 T_2 \right ) +n \\ &\geq n - {\rm nul} (T_1) + k - {\rm nul} (T_2) \\ &={\rm rank} (T_1) + {\rm rank} (T_2) \end{align*}

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