Let A be a m n matrix and B a n p matrix Suppose that the

Let A be a m × n matrix and B a n × p matrix. Suppose that the m × p matrix C = AB has the property that Cx = 0 has only the trivial solution. Find dim Col(B), the dimension of the column space of B. Be sure to justify your answer.

Solution

If rank( C)= r, the solutions of Cx=0 is mr dimensional space, so if Cx = has only trivial solutions it means that rank(C) = size( C) so C is invertible i.e. m = p and det ( C) 0. Also then, A is a m x n matrix, B is a n x m matrix and C is a m x m matrix. Further, since det ( C ) = det (AB) = det(A) det(B), therefore det (A)0 and det (B)0. We know that whenever some column of a matrix can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), its determinant is 0. Since det (B)0, the columns of B are linearly independent. Therefor, the dimension of Col (B) is m.