## Linear Algebra and Its Applications, Exercise 1.5.1

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 $a_{ij} \ne 0$ 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 .

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