Guided Proof Prove that a triangular matrix isnonsingular

Chapter 7, Problem 7.59

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