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Exercise 8.9.1

Avg difficulty 3.0

Find the algebraic and geometric multiplicity of each eigenvalue of the following matrices. Which of the matrices are diagonalizable?

(a)A=[11−13],(b)B=[−78−45],(a)\quad A=\left[\begin{array}{rr}1&1\\ -1&3\end{array}\right],\quad(b)\quad B=\left[\begin{array}{rr}-7&8\\ -4&5\end{array}\right], (c)C=[2−1−10420−11],(d)D=[31−10210−14],(e)E=[−201−1−11−210].(c)\quad C=\left[\begin{array}{rrr}2&-1&-1\\ 0&4&2\\ 0&-1&1\\ \end{array}\right],\quad(d)\quad D=\left[\begin{array}{rrr}3&1&-1\\ 0&2&1\\ 0&-1&4\\ \end{array}\right],\quad(e)\quad E=\left[\begin{array}{rrr}-2&0&1\\ -1&-1&1\\ -2&1&0\\ \end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Eigenvalues, eigenvectors, and diagonalization, Properties of eigenvectors and eigenvalues, Exercise 8.9.1

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.