Matrix Multiplication and Rank of Matrix

[rajtendulkar]

Dear Forum,

I have one question on matrix multiplication.

Suppose there are 2 matrices -

A = 1 -1 0
0 2 -1
2 0 -1

B = 1
1
2

and AB = 0 (Zero Matrix)
if B not a zero-matrix, then rank(A) is less than s, where s is the dimension of B.

I wanted to have a proof and explanation for this.
Any book / link? :)

Thank You,
Raj

 

[spamiam]

Welcome to PF, Raj!

Check out this link: http://en.wikipedia.org/wiki/Rank–nullity_theorem. Then note that b is a nonzero element of the null space of A.

Hope that was clear!

 

[rajtendulkar]

Thanks a lot for the reply ! :)