# When an m by n matrix A has rank r = m, the system Ax = b can be solved for which b(best answer)? How many special solutions to Ax = O?

I've been struggling with this question from Linear Algebra - Step by Step by KULDEEP SINGH, Oxford University Press.

When an m by n matrix A has rank r = m, the system Ax = b can be solved for which b(best answer)? How many special solutions to Ax = O?

Image for the whole question

Could you please help? I can't make sense of the last question (c) below. What does it mean by "for which b"?

I guess it asks for the condition for which the system Ax = b has a unique solution, when the Augmented matrix Axb has rank = m, and m=n, or an infinite number of solutions when the Augmented matrix Axb has rank = m and m<n. Is it correct?

For the second part of the question, as I understand it, is it asking me to explain when the homogeneous system will have a unique solution and when infinitely many solutions?

Thank you very much.

• Please don't use pictures, use MathJax. Here is a tutorial. Jun 18 at 8:15