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Recall that the characteristic polynomial for a 2 x 2 matrix A is P(X) = det(A- AI). (a) Explain why the solutions to P(X) = 0 are the eigenvalues of A. (b) If A1 and A2 are the eigenvalues of A, explain why P(A) = (A – Ai)(A – A2). (c) Prove that det(A) = A12.

Recall that the characteristic polynomial for a 2 x 2 matrix A is P(X) = det(A- AI). (a) Explain why the solutions to P(X) = 0 are the eigenvalues of A. (b) If A1 and A2 are the eigenvalues of A, explain why P(A) = (A – Ai)(A – A2). (c) Prove that det(A) = A12.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 41EQ
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Recall that the characteristic polynomial for a 2 x 2 matrix A is P(X) = det(A- AI). (a) Explain why the solutions to P(X) = 0 are the eigenvalues of A. (b) If A1 and A2 are the eigenvalues of A, explain why P(X) = ( Xi)(A – A2). (c) Prove that det(A) = A12. (d) Explain why if det(A) = 0, then the matrix must have at least one eigenvalue equal to zero. (e) Interpret geometrically what it means for a matrix A to have A1 0 (with eigenvector vi).
Transcribed Image Text:Recall that the characteristic polynomial for a 2 x 2 matrix A is P(X) = det(A- AI). (a) Explain why the solutions to P(X) = 0 are the eigenvalues of A. (b) If A1 and A2 are the eigenvalues of A, explain why P(X) = ( Xi)(A – A2). (c) Prove that det(A) = A12. (d) Explain why if det(A) = 0, then the matrix must have at least one eigenvalue equal to zero. (e) Interpret geometrically what it means for a matrix A to have A1 0 (with eigenvector vi).
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