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Math Help - Rank of partitioned matrix

  1. #1
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    Rank of partitioned matrix

    Given a 2n x 2n matrix M which can be decomposed into to matrices M = [A B] where A and B are 2n x n matrices. Rank(A) = n, Rank(B) = n and also ANY n rows of A are linearly independent and also any n rows of B. Also, any n rows of the 4n x n matrix
    [A]
    B]
    are linearly independent.

    Is it true that Rank(M) = 2n and how can it be shown?
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  2. #2
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    first idea...

    Here's now an idea how I could show it -- but actually I think, this is not totally correct

    Let's look at the rows of (A B). Since all rows of A are linearly independent, we need a linear combination of n+1 rows of M such that we obtain a vector
    (0 v_2).

    v_2 is unequal 0 at all entries since the rows of B are linearly independent of the rows of A. (IS THIS TRUE?)

    There exist further r-1 rows of B which are linearly independent of (0 v_2).

    Hence rank(A B) = 2n.

    Is this proof correct?? I am not happy with this proof at all...
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