Show that rank AB LE min rankA rank B Also show that rank AB

Show that rank (AB) LE min (rank(A), rank (B)). Also show that rank (AB) = rank (AB) if B is nonsingular.

Solution

We know linear transformation of AB is same as linear transformation A o linear transformation of B

But we know rank(AoB) <= rank(A) and rank(AoB) <= rank(B)

Therefore rank(AoB) <= min(rank(A),rank(B))

If B is a non singular matrix, its rank would be size of the matrix

we know that

rank(AoB) <= min(rank(A),rank(B))

Hence if matrix is non singular, minimum vaue would depend on A.

Hence rank(AB) = rank(A) if B is non singular