Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**4.3.1. Determine which of the following matrices are (i) orthogonal; (ii) proper orthogonal.**
Given Matrices:
1.
\[
\begin{pmatrix}
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6}
\end{pmatrix}
\]
2.
\[
\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & -1
\end{pmatrix}
\]
3.
\[
\begin{pmatrix}
-\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3} \\
\frac{2}{3} & \frac{2}{3} & -\frac{1}{3}
\end{pmatrix}
\]
The task is to determine for each matrix if it is orthogonal or proper orthogonal. An orthogonal matrix has its transpose equal to its inverse, and a proper orthogonal matrix also has a determinant of +1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31065e4-6071-4dbc-99a3-4d8b8e7c0b94%2F8e2999aa-870f-4c58-ba3b-82c0c26de926%2Fswxull_processed.png&w=3840&q=75)
Transcribed Image Text:**4.3.1. Determine which of the following matrices are (i) orthogonal; (ii) proper orthogonal.**
Given Matrices:
1.
\[
\begin{pmatrix}
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
\frac{1}{4} & \frac{1}{5} & \frac{1}{6}
\end{pmatrix}
\]
2.
\[
\begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & -1
\end{pmatrix}
\]
3.
\[
\begin{pmatrix}
-\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3} \\
\frac{2}{3} & \frac{2}{3} & -\frac{1}{3}
\end{pmatrix}
\]
The task is to determine for each matrix if it is orthogonal or proper orthogonal. An orthogonal matrix has its transpose equal to its inverse, and a proper orthogonal matrix also has a determinant of +1.
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