a problem regarding rank of matrices

Jan 2009
9
0
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
Mar 2010
3,093
582
Florida
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 \(\displaystyle n\leq m\)?