). Expand The equation,2,3.and SimbewhenAij=Aje(a,Symmetnic(Ante-

Answered: ). Expand The equation ,2, 3. and Simbe…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

). Expand The equation
,2,
3.
and Simbe
when
Aij=Aje
(a,
Symmetnic
(Ante-

Expert Solution

Step 1

(a) Given $A_{i j} = A_{j i} (S y m m e t r i c)$

We have to expand $A_{i j} x_{i} x_{j}$ and then simplify it

$\begin{matrix} A_{i j} x_{i} x_{j} = A_{11} x_{1} x_{1} + A_{12} x_{1} x_{2} + A_{13} x_{1} x_{3} + A_{21} x_{2} x_{1} + A_{22} x_{2} x_{2} + A_{23} x_{2} x_{3} + A_{31} x_{3} x_{1} + A_{32} x_{3} x_{2} + A_{33} x_{3} x_{3} \\ = A_{11} x_{1} x_{1} + A_{12} x_{1} x_{2} + A_{13} x_{1} x_{3} + A_{12} x_{2} x_{1} + A_{22} x_{2} x_{2} + A_{23} x_{2} x_{3} + A_{13} x_{3} x_{1} + A_{23} x_{3} x_{2} + A_{33} x_{3} x_{3} (\begin{array}{l} Use A_{21} = A_{12}, \\ A_{31} = A_{13} \\ and A_{32} = A_{23} \end{array}) \\ = A_{11} x_{1} x_{1} + A_{22} x_{2} x_{2} + A_{33} x_{3} x_{3} + 2 A_{12} x_{1} x_{2} + 2 A_{13} x_{1} x_{3} + 2 A_{23} x_{2} x_{3} \\ = A_{11} x_{1} x_{1} + A_{22} x_{2} x_{2} + A_{33} x_{3} x_{3} + 2 (A_{12} x_{1} x_{2} + A_{13} x_{1} x_{3} + A_{23} x_{2} x_{3}) \end{matrix}$

Hence, $A_{i j} x_{i} x_{j} = A_{11} x_{1} x_{1} + A_{22} x_{2} x_{2} + A_{33} x_{3} x_{3} + 2 (A_{12} x_{1} x_{2} + A_{13} x_{1} x_{3} + A_{23} x_{2} x_{3})$