3. Let A be a n x n matrix. Suppose that A 7 Onn, A # In, and A? = A. (A) Show that A has exactly 2 eigenvalues: 0, 1. (B) Let v E nullA. Show that v is an eigenvector of A with eigenvalue 0. (C), Let v E colA. Show that v is an eigenvector of A with eigenvalue 1. (D) Show that A is diagonalizable.

3. Let A be a n x n matrix. Suppose that A 7 Onn, A # In, and A? = A. (A) Show that A has exactly 2 eigenvalues: 0, 1. (B) Let v E nullA. Show that v is an eigenvector of A with eigenvalue 0. (C), Let v E colA. Show that v is an eigenvector of A with eigenvalue 1. (D) Show that A is diagonalizable.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let A be a n x n matrix. Suppose that A 7 0nn, A + In, and A? = A.
(A) Show that A has exactly 2 eigenvalues: 0, 1.
(B) Let v E nullA. Show that v is an eigenvector of A with eigenvalue 0.
(C), Let v E colA. Show that v is an eigenvector of A with eigenvalue 1.
(D) Show that A is diagonalizable.
3.

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