Question

A set of linear equation is represented by the matrix equation AX = B. The necessary condition for the existence of a solution for this system is

• A. A must be invertible
• B. B must be linearly dependent of columns of A
• C. B must be linearly independent on the columns of A
• D. None

Answer 1

The correct option is B B must be linearly dependent of columns of A

Consider $A_{3x3}$ then $[A:B{]}_{3x4}$, i.e., Augmented mayrix have 4 vectors. But by property of rank, we have $\rho (A:B)\le 3$ which is definitely less than 4 so [A:B] has LD set of vectors so we can conclude that martrix [B] must be linearly dependent of columns of matrix [A].