Asked by Renuka Ganesh | 22nd May, 2014, 02:43: PM

Expert Answer:

N o w space c o n s i d e r space t h e space m a t r i x space open square brackets table row 1 a b row omega 1 c row cell omega squared end cell omega 1 end table close square brackets G i v e n space t h a t space a comma space b comma space a n d space c space i s space e i t h e r space omega space o r space omega squared. I f space w e space A s s u m e space a equals omega squared comma space t h e n space t h e space m a t r i x space w o u l d space b e space sin g u l a r. T h u s comma space t h e space v a l u e space o f space a space s h o u l d space b e space e q u a l space t o space omega. L e t space u s space c o n s i d e r space t h e space v a l u e space o f space c space t o space b e space e q u a l space t o space omega squared. A g a i n space t h i s space l e a d s space t o space a space sin g u l a r space m a t r i x space a n d space h e n c e space v a l u e space c space i s space a l s o space e q u a l space t o space omega. rightwards double arrow a equals c equals omega. N o w space c o n s i d e r space t h e space v a l u e space o f space b space a s space omega. T h u s comma space t h e space v a l u e space o f space t h e space d e t e r m i n a n t space space open vertical bar table row 1 omega omega row omega 1 omega row cell omega squared end cell omega 1 end table close vertical bar equals 1 cross times open parentheses 1 minus omega squared close parentheses minus omega cross times open parentheses omega minus omega cubed close parentheses plus omega cross times open parentheses omega squared minus omega squared close parentheses equals 1 minus omega squared minus omega cross times open parentheses omega minus 1 close parentheses equals 1 minus omega squared minus omega squared plus omega equals 1 minus 2 omega squared plus omega not equal to 0 A l s o space n o t e space t h a t space t h e space v a l u e space o f space t h e space d e t e r m i n a n t space i s space i n d e p e n d e n t space o f space b. S o space b space c a n space a s s u m e space a n y space o f space t h e space t w o space g i v e n space v a l u e s. S o space t h e space o n l y space w a y space t h e space m a t r i x space i s space n o n minus sin g u l a r space i s space w h e n space a equals c equals omega. S i n c e space b space c a n space h a v e space e i t h e r space o f space t h e space t w o space v a l u e s comma space t h e r e space a r e space n o n minus sin g u l a r space m a t r i c e s space o f space t h e space g i v e n space t y p e.
S o space a n s w e r space i s space 2 space a n d space o p t i o n space A space i s space c o r r e c t.

Answered by Vimala Ramamurthy | 23rd May, 2014, 03:14: PM

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