Develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.

Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.) 

a. Show that if B is n × p, then rank AB ≤ rank A.

b. Show that if B is n × p, then rank AB  ≤ rank B.