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Chapter 3: Q3P (page 158)

(a) If Cis orthogonal and Mis symmetric, show that C-1MCis symmetric.

(b) IfC is orthogonal and Mantisymmetric, show thatC-1MCis antisymmetric.

Short Answer

Expert verified

a)C-1MCis symmetric.

b) C-1MCis antisymmetric.

Step by step solution

01

Given information from question

a) It is given that, C is orthogonal, and M is symmetric.

b) It is given that, C is orthogonal, and M is antisymmetric.

02

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

03

Calculate C-1MC is symmetric

a)

We have givenis orthogonal andis symmetric.

So,

C-1MCT=CT-1MTCT=C-1-1MTCT=CMC-1

Where CT=C-1since C is orthogonal. And MT=Mwhere M is symmetric

04

Calculate C-1MC is antisymmetric

MT=M(b)

Since we have givenis orthogonal andis antisymmetric.

So,

C-1MCT=CT-1MTCT=C-1-1MTCT=C-MC-1=-CMC-1

Where CT=C-1since C is orthogonal and MT=Mwhere M is antisymmetric.

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