[4] a-Find anon zero 3x3 matrix A such that A" = -A.-3b- Find A , if ( 5 A")-1 = ||:52

Answered: [4] a-Find anon zero 3x3 matrix A such…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.6: Matrices

Problem 11E: 11. Find two nonzero matrices and such that.

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[4] a-Find anon zero 3x3 matrix A such that A" = -A.
-3
b- Find A , if ( 5 A")-1 = ||
:
5
2

Expert Solution

Step 1

According to part (a) of question

we have to find non zero 3*3 matrix A such that $A^{T} = - A$ .

According to part (b) of question

we have to find value of A

Step 2

(a)let $A = |\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}|$ $\Rightarrow A^{T} = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}]$

given that

$A^{T} = - A [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}] = - [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] \Rightarrow b = - d, c = - g, f = - h, a = - a, e = - e, i = - i f r o m w h i c h a = - a, e = - e a n d i = - i i s o n l y p o s s i b l e w h e n a = 0, e = 0 a n d i = 0 h e n c e u \sin g a b o v e v a l u e s i n m a t r i x a w e g e t [\begin{matrix} 0 & - d & - g \\ d & 0 & - h \\ g & h & 0 \end{matrix}] h e r e w e g e t t h e r e q u i r e d m a t r i x A h e r e w e c a n p u t a n y v a l u e i n d, g a n d h f o r e x a m p l e l e t d = 3, g = 5 a n d h = 7 t h e n A = [\begin{matrix} 0 & 3 & 5 \\ - 3 & 0 & 7 \\ - 5 & - 7 & 0 \end{matrix}]$

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