Prove that if the product of two matrices is the zero matrix

Prove that if the product of two matrices is the zero matrix, then at least one of the factors has determinant 0.

Solution

Let the two matrices be A,B

then AB=0

we know that det(AB)=det(A)*det(B)

then 0=det(A)*det(B)

so det(A)=0 or det(B)=0