Question: 1. From Theorem 7.1 in the study guide (and Corollary 1), we know that if we have n distinct eigenvalues for a n n matrix,

1. From Theorem 7.1 in the study guide (and Corollary 1), we know that if we have n distinct eigenvalues for a n n matrix, then we will be able to find linearly independent eigenvectors and hence diagonalise the matrix. i. For a 2 2 matrix , a b c d ,algebraically determine conditions that would lead to (a) distinct eigenvalues, (b) a repeated eigenvalue with only one non-linearly independent vector, and (c) a repeated eigenvalue with two linearly independent vectors. ii. Hence provide original examples of each type. iii. Can you find any similar conditions for the different cases for a 3 3 matrix? 10 marks

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