Exercise 1.5.1. Given an upper triangular matrix A, under what conditions is A nonsingular?
Answer: If A is upper triangular then the diagonal entries of A are the pivots of the corresponding system of linear equations. For A to be nonsingular then all the pivots must be nonzero. (If even one pivot is zero then A is singular, since A is already in upper triangular form and thus the zero pivot cannot be rectified using a row exchange.)
So A is nonsingular if and only if for all i, j.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.