Question

it's given A matrix and B matrix different from 0 that and it's real that AB=0 PROVE THAT THESE MATRIX ARE BOTH DEGENERATED

Answer 1

You mean if A and B are non-zero matrices, then
AB = 0 => both A and B are degenerate.

Answer 2

YES

Answer 3

Well, it's clear that at least one of them must be degenerate because

AB = 0 => det(AB) = det(0) = 0 => det(A) det(B) = 0
and hence at least one of det(A) and det(B) must be zero.

Answer 4

What's less clear is that both of them must be degenerate. To tackle that I'd look at
Ab, where b is a column vector from B
and aB = (B^t a^t)^t where a is a row vector in A.

Each one of those products must be the zero column vector. Now from that you can conclude something.

Answer 5

zero column vector or zero row vector respectively

Answer 6

YES I UNDERSTAND THAT BUT HERE IT'S WRITTEN THAT WE MUST PROVE THAT BOTH ARE DEGENERATED

Answer 7

Please don't write in block caps. It makes you look like you're shouting.

Second, "degenerated" is not the standard word here, but it doesn't matter; we're all working towards to the same thing.

Third, given the approach I just laid out, what you'll need is the fact both A and B are not the zero matrix. Which means that the span of their column space/row space is at least one-dimensional; i.e., the rank of each matrix is at least one.

And now that means that the kernel or null space of A and B is at least 1-dimensional. This gives you what you need if you know the definition of 'degenerate' ('degenerated') and what it's equivalent to.

Answer 8

ok sorry i didn't want to look like that and thank you