if the columns of B are linearly dependent then the columns

if the columns of B are linearly dependent, then the columns of AB are also linearly independent

Solution

Columns of B being linearly dependent means that there is a nonzero vector x such that Bx = 0.

Or we can say that when the columns of B are linearly dependent, then the determinant of B is zero.

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

which means  columns of AB must also be linearly dependent.