Would someone please show me how to show that:

$\displaystyle rank(A) + rank(B) \leq rank(C) $

Thank you.

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- Jan 8th 2012, 03:38 PMitproRank proof
Would someone please show me how to show that:

$\displaystyle rank(A) + rank(B) \leq rank(C) $

Thank you. - Jan 8th 2012, 04:02 PMalexmahoneRe: Rank proof
- Jan 8th 2012, 04:07 PMitproRe: Rank proof
- Jan 8th 2012, 04:08 PMalexmahoneRe: Rank proof
- Jan 8th 2012, 04:13 PMitproRe: Rank proof
- Jan 9th 2012, 03:30 AMDevenoRe: Rank proof
what you want to show is that:

rank(A) + rank(B) ≤ rank(A+B).

but this is not true. suppose that A =

[1 0]

[0 0] and that B =

[-1 0]

[ 0 0].

clearly, A and B both have rank 1, so rank(A) + rank(B) = 1 + 1 = 2.

however, A+B is the 0-matrix, which has rank 0, and 2 is not less than or equal to 0.