AI Tutor
Back to Diagonalization, similarity, and powers of a matrix

Activity 4.3.3

Avg difficulty 2.5

  1. Find a diagonalization of AA, if one exists, when A=[3−26−5]A = \left[\begin{array}{rr} 3 & -2 \\ 6 & -5 \\ \end{array}\right].

  2. Can the diagonal matrix A=[200−5]A = \left[\begin{array}{rr} 2 & 0 \\ 0 & -5 \\ \end{array}\right] be diagonalized? If so, explain how to find the matrices PP and DD.

  3. Find a diagonalization of AA, if one exists, when A=[−2001−3020−3]A = \left[\begin{array}{rrr} -2 & 0 & 0 \\ 1 & -3& 0 \\ 2 & 0 & -3 \\ \end{array}\right].

  4. Find a diagonalization of AA, if one exists, when A=[−2001−3021−3]A = \left[\begin{array}{rrr} -2 & 0 & 0 \\ 1 & -3& 0 \\ 2 & 1 & -3 \\ \end{array}\right].

  5. Suppose that A=PDP−1A=PDP^{-1} where D=[300−1],P=[v2v1]=[221−1]D = \left[\begin{array}{rr} 3 & 0 \\ 0 & -1 \\ \end{array}\right],\qquad P = \left[\begin{array}{cc} \vvec_2 & \vvec_1 \end{array}\right] = \left[\begin{array}{rr} 2 & 2 \\ 1 & -1 \\ \end{array}\right].

    1. Explain why AA is invertible.

    2. Find a diagonalization of A−1A^{-1}.

    3. Find a diagonalization of A3A^3.

Austin, Understanding Linear Algebra, §4.3 Diagonalization, similarity, and powers of a matrix, Activity 4.3.3

AI Tutor

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