1. Decide if each of the following statements is true or false and prove it. Not

Question

1. Decide if each of the following statements is true or false and prove it. Note: Two matrices A and B are similar if A = SBS^-1 for some matrix S. (a) Every n x n matrix has n distinct eigenvalues. (b) If a matrix has one eigenvector, then it has infinitely many eigenvectors. (c) There exists a square matrix with real entries having no real eigenvalues. (d) There exists a square matrix with no eigenvectors. (e) The sum of two eigenvectors of a matrix is always an eigenvector. (f) Similar matrices have the same eigenvalues. Hint: You can prove that det(A ? lambda I) = det(B ? lambdal) if A and B are similar.

Explanation / Answer

a. false

b. true

c. true

d. false

e. true

f. true

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