1. The problem statement, all variables and given/known data Problem: Consider a system of linear equations in echelon form with r equations and n unknowns. Prove the following.: (i) If r = n, then the system has a unique solution. (ii) If r < n, then we can arbitrarily assign values to the n - r free variables and solve uniquely for the r pivot variables, obtaining a solution of the system. Solution: (i) Suppose r = n. Then we have a square system AX = B where the matrix A of coefficient sis (upper) triangular with nonzero diagonal elements. Thus, A is invertible. By Theorem 3.10, the system has a unique solution. (ii) Assigning values to the n - r free variables yields a triangular system in the pivot variables, which, by (i), has a unique solution. Statement of Theorem 3.10: Suppose the field K is infinite. Then the system AX = B has: (a) a unique solution, (b) no solution, or (c) an infinite number of solutions. 2. Relevant equations There is a theorem (that I found online) which states than an upper triangular matrix is invertible if and only if all of its diagonal elements are non zero. 3. The attempt at a solution I understand the solution for part (i) up until and including "Thus, A is invertible.", but I don't get the part that says "By Theorem 3.10, the system has a unique solution.". How does one come to that conclusion from Theorem 3.10? In other words, how does one determine from Theorem 3.10 that the system has a unique solution as opposed to having no solution or an infinite number of solutions? I'm not asking for a general way to justify determining that the system has a unique solution; I'm asking for a way to justify that the system has a unique solution by specifically using Theorem 3.10. If something I said is unclear, let me know. Any help in understanding the solution of this proof would be greatly appreciated!