Question

If r is the rank of the m × n matrix A then the number of linearly independent solutions of AX=0 is

Answer 1

Step-by-step explanation:

I came across the following multiple choice question:

The number of linearly independent solution of the homogeneous system of linear equations AX=0, where X consists of n unknowns and A consists of m linearly independent rows is

(A) m−n (B) m (C) n−m (D) none of these

I think the answer will be (D) because:

When m=n, in this case it would mean a square matrix with all linearly independent rows, which implies unique solution. When m<n, it would mean that the rank of the matrix is less than number of unknowns in the system, which would mean infinite solutions and these solutions must be linearly dependent (Am I going right?). When m>n, there will be no solution (I am not sure about this one)

I think there is something wrong about my answer because I remember something like: number of linearly independent solutions = number of unknowns - rank, from my Linear Algebra class. But I am not sure how to relate to it here..

Thanks..

Answer 2

When m=n, in this case it would mean a square matrix with all linearly independent rows, which implies unique solution. When m<n, it would mean that the rank of the matrix is less than number of unknowns in the system, which would mean infinite solutions and these solutions must be linearly dependent . When m>n, there will be no solution