Linear Algebra Suppose that A is a 3 5 matrix and that B is

Linear Algebra:

Suppose that A is a 3 × 5 matrix and that B is a 4 × 3 matrix — so that the product BA is a 4 × 5 matrix.

a) If rank(A) = 2 and rank(B) = 3, what are the minimum and maximum possible ranks of BA?

b) If rank(A) = 2 and rank(B) = 2, what are the minimum and maximum possible ranks of BA.

c) If you know nothing about the ranks of A and B, what are the minimum and maximum possible ranks of BA?

Solution

rank(BA)<=min{rank(A),rank(B)}

a)rank(BA)<=2

maximum possible rank=2

minimum possible rank=0

b)rank(BA)<=2

maximum possible rank=2

minimum possible rank=0

c)

maximum possible rank=min{rank(A),rank(B)}

minimum possible rank=0