asst3 - MATH 235, Fall 2007 Assignment #3 1. Let A be an n...

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MATH 235, Fall 2007 Assignment #3 1. Let A be an n × n matrix. Suppose that λ 1 ,...λ k are the (distinct) eigenvalues of A and that their multiplicites are m λ 1 ,...,m λ k , respectively. Then, prove that det ( A ) = k Y i =1 λ m λ i i . 2. Let A be a 2 × 2 matrix. Prove that the characteristic polynomial of A is λ 2 - trace ( A ) λ + det ( A ) . 3. Let A be a 5 × 5 matrix. Suppose we are given the following facts about A : trace ( A ) = 10, 2 is an eigenvalue of A with multiplicity 3, and A is not invertible. Prove that this information completely determines the characteristic polynomial of A , and state the characteristic polynomial of A . 4. Let A be a square matrix and let x,y and z be three eigenvectors of A . Suppose that x and y both correspond to the same eigenvalue λ and are linearly indepen- dent, while z corresponds to a different eigenvalue μ . Prove that { x,y,z } is a linearly independent set. 5. For each of the following matrices
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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