Final answer:
The question revolves around the condition for a square matrix A being invertible, which is evident when the only solution to Ax = 0 is the trivial solution, signifying that Ax = b will have a unique solution for any n × 1 matrix b.
Step-by-step explanation:
The question discusses a fundamental concept in linear algebra that involves solutions to the system of linear equations represented by the matrix equation Ax = b. According to the statement, the matrix equation Ax = 0 has only the trivial solution (the zero vector) if and only if the matrix equation Ax = b has a unique solution for every possible n × 1 matrix b. This relates to the invertibility of matrix A, which implies that A is a square matrix with a non-zero determinant, thus having full rank. This ensures that the system of equations has unique solutions. It is a necessary and sufficient condition for A to be invertible that the homogeneous equation Ax = 0 has only the trivial solution. If there are non-trivial solutions to this equation, then A does not have full rank, is not invertible, and correspondingly, the equation Ax = b can either have no solutions or an infinite number of solutions for some b.