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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Determine whether the matrix is symmetric, orthogonal, both, or neither.

Expert Solution

Step 1

The given matrix is

$A = [\begin{matrix} \frac{1}{3} & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{2} & \frac{1}{3} \end{matrix}]$

We have to check whether given matrix is symmetric, orthogonal or not.

Step 2

The matrix A is symmetric if $A = A^{T}$ .

The matrix A is an orthogonal, if $A A^{T} = A^{T} A = I$ .

We have

$A = [\begin{matrix} \frac{1}{3} & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{2} & \frac{1}{3} \end{matrix}]$

Now, the transpose of the matrix A is

$A^{T} = [\begin{matrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{matrix}]$

$A \neq A^{T}$

This implies A is not symmetric.

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