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Back to Diagonalization, similarity, and powers of a matrix

Exercise 1

Avg difficulty 2.5

Determine whether the following matrices are diagonalizable. If so, find matrices DD and PP such that A=PDP−1A=PDP^{-1}.

  1. A=[−2−2−21]A = \left[\begin{array}{rr} -2 & -2 \\ -2 & 1 \\ \end{array}\right].

  2. A=[−11−1−3]A = \left[\begin{array}{rr} -1 & 1 \\ -1 & -3 \\ \end{array}\right].

  3. A=[3−42−1]A = \left[\begin{array}{rr} 3 & -4 \\ 2 & -1 \\ \end{array}\right].

  4. A=[1002−20014]A = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 2 & -2 & 0 \\ 0 & 1 & 4 \\ \end{array}\right].

  5. A=[122212221]A = \left[\begin{array}{rrr} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\ \end{array}\right].

Austin, Understanding Linear Algebra, §4.3 Diagonalization, similarity, and powers of a matrix, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.