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Exercise 11.6.2

Avg difficulty 3.0

Which of the following matrices define orthogonal transformations? Which ones define isometries?

A=[100−1],B=12[11−11],C=15[1−221],D=14[12222−1].A=\left[\begin{array}{rr}1&0\\ 0&-1\\ \end{array}\right],\quad B=\frac{1}{2}\left[\begin{array}{rr}1&1\\ -1&1\\ \end{array}\right],\quad C=\frac{1}{\sqrt{5}}\left[\begin{array}{rr}1&-2\\ 2&1\\ \end{array}\right],\quad D=\frac{1}{4}\left[\begin{array}{rr}1&2\\ 2&2\\ 2&-1\\ \end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonal functions and orthogonal matrices, Exercise 11.6.2

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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.