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Exercise 11.6.3

Avg difficulty 3.0

Determine which of the following matrices are orthogonal by checking whether the columns form an orthonormal set.

A=13[12221−2−22−1],B=[002020200],C=13[2−12−12−22−21],D=15[3404−30005].A=\frac{1}{3}\left[\begin{array}{rrr}1&2&2\\ 2&1&-2\\ -2&2&-1\\ \end{array}\right],\quad B=\left[\begin{array}{rrr}0&0&2\\ 0&2&0\\ 2&0&0\\ \end{array}\right],\quad C=\frac{1}{3}\left[\begin{array}{rrr}2&-1&2\\ -1&2&-2\\ 2&-2&1\\ \end{array}\right],\quad D=\frac{1}{5}\left[\begin{array}{rrr}3&4&0\\ 4&-3&0\\ 0&0&5\\ \end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonal functions and orthogonal matrices, Exercise 11.6.3

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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.