Guided Proof Prove that a triangular matrix is nonsingular if and only if its

Chapter 7, Problem 57

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Guided Proof Prove that a triangular matrix is nonsingular if and only if its eigenvalues are real and nonzero. Getting Started: Because this is an if and only if statement, you must prove that the statement is true in both directions. Review Theorems 3.2 and 3.7. (i) To prove the statement in one direction, assume that the triangular matrix is nonsingular. Use your knowledge of nonsingular and triangular matrices and determinants to conclude that the entries on the main diagonal of are nonzero. (ii) Because is triangular, you can use Theorem 7.3 and part (i) to conclude that the eigenvalues are real and nonzero. (iii) To prove the statement in the other direction, assume that the eigenvalues of the triangular matrix are real and nonzero. Repeat parts (i) and (ii) in reverse order to prove that is nonsingular.