# a problem regarding rank of matrices

#### dull1234

Hey guys,
can anyone help me with this problem please ??

Given A be m * n (m by n) matrix and B be n * p (n by p) matrix

Show that 1. rank(AB) is less than or equal to rank(A)
2. rank(AB) is less than or equal to rank(B)
3. If B is invertible, than rank(AB) = rank(A)

i think 1 and 2 are kind of similar ? so clear explanation on 1 and 3 are appreciated. Thanks so much!!

#### dwsmith

MHF Hall of Honor
Hey guys,
can anyone help me with this problem please ??

Given A be m * n (m by n) matrix and B be n * p (n by p) matrix

Show that 1. rank(AB) is less than or equal to rank(A)
2. rank(AB) is less than or equal to rank(B)
3. If B is invertible, than rank(AB) = rank(A)

i think 1 and 2 are kind of similar ? so clear explanation on 1 and 3 are appreciated. Thanks so much!!
The rank of A can at most be m, the rank of B can at most be n.

$$\displaystyle AB=C_{m x p}$$; therefore, the rank of C can at most be m.

Do we know if $$\displaystyle n\leq m$$?