1. verify that (a) the sum of the eigenvalues of A equal the trace of A and (b)

Question

1. verify that (a) the sum of the eigenvalues of A equal the trace of A and (b) the sum of the eigenvalues of A equals determine of A, where A=(3 2 -3

-3 -4 9

-1 -2 5)

2. Cayley-Hamilton theorem states that a matrix satisfies its characteristic polynomial. that is , if p^a is the characteristic polnomial of a square matrix A, then p^a(A)=0. verify cayley-hamilton theorem for A= ( 1 4

3 11)

3.find an invertible matrix P such that P^-1 AP is a diagonal matrix

where A=(1 -1 2

-1 0 1

-1 -2 3)

4.If A is an n-by-n diagonalizable matrix, then find rank(A)

5. suppose the a 7-by-7 diagonalizable matrix has 3 distincteigenvalues, one with eigenspace of dimension 1 and another with eigenspace of dimension 2. What is the dimension of the 3rd eigenspace? Explain

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