Question

8. Prove the Cayley-Hamilton Theorem: If A ? M3 3(R) and p(t) is its characteristic polynomial, then p(A) = 0 (the 3 3 zero matrix.) 9. a) Show that a matrix A ? M2 2(R) has distinct real eigenvalues if and only if (Tr(A)) 2 > 4 det(A). b) Show that A has a real eigenvalue of multiplicity 2 if and only if (Tr(A)) 2 = 4 det(A). c) What happens if (Tr(A)) 2 < 4 det(A)? 10. a) Show that a matrix A ? M3 3(R) always has at least one real eigenvalue. Give an example of such a matrix that has only one real eigenvalue, with multiplicity 1. b) How can this result be generalized? 11. a) Show that if A = [a11 ... a1n; 0 a22 ...; ... ... ...; 0 ... 0 ann] is an "upper-triangular" n x n matrix (i.e. all entries below the main diagonal are zero), then the eigenvalues of A are the entries on the main diagonal.

          8. Prove the Cayley-Hamilton Theorem: If A ? M3  3(R) and p(t) is its characteristic polynomial, then p(A) = 0 (the 3   3 zero matrix.)
9. a) Show that a matrix A ? M2  2(R) has distinct real eigenvalues if and only if (Tr(A))  2 > 4    det(A).
b) Show that A has a real eigenvalue of multiplicity 2 if and only if (Tr(A))  2 = 4    det(A).
c) What happens if (Tr(A))  2 < 4    det(A)?
10. a) Show that a matrix A ? M3  3(R) always has at least one real eigenvalue. Give an example of such a matrix that has only one real eigenvalue, with multiplicity 1.
b) How can this result be generalized?
11. a) Show that if A = [a11 ... a1n; 0 a22 ...; ... ... ...; 0 ... 0 ann] is an "upper-triangular" n x n matrix (i.e. all entries below the main diagonal are zero), then the eigenvalues of A are the entries on the main diagonal.

8. Prove the Cayley-Hamilton Theorem: If A ? M3 3(R) and p(t) is its characteristic polynomial, then p(A) = 0 (the 3 3 zero matrix.)
9. a) Show that a matrix A ? M2 2(R) has distinct real eigenvalues if and only if (Tr(A)) 2 > 4 det(A).
b) Show that A has a real eigenvalue of multiplicity 2 if and only if (Tr(A)) 2 = 4 det(A).
c) What happens if (Tr(A)) 2 < 4 det(A)?
10. a) Show that a matrix A ? M3 3(R) always has at least one real eigenvalue. Give an example of such a matrix that has only one real eigenvalue, with multiplicity 1.
b) How can this result be generalized?
11. a) Show that if A = [a11 ... a1n; 0 a22 ...; ... ... ...; 0 ... 0 ann] is an "upper-triangular" n x n matrix (i.e. all entries below the main diagonal are zero), then the eigenvalues of A are the entries on the main diagonal.

Added by Ashley C.

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