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Question

If D = diag (d1, d2,.....,dn), then Dn equals -
 

  1. D    
  2. diag (d1n–1, d2n–1, ....dnn–1)    
  3. diag (d1n, d2n, .....,dnn)    
  4. None of these    

The correct answer is: diag (d1n, d2n, .....,dnn)


    we are given that  D = diag (d1, d2,.....,dn) and we need to find what  Dn equals -
    A space d i a g o n a l space m a t r i x space i s space a space s q u a r e space m a t r i x space i n space w h i c h space t h e space e n t r i e s space o u t s i d e space t h e space m a i n space d i a g o n a l space a r e space a l l space z e r o s.
T h e r e f o r e space t h e space d e t e r m i n a n t s space o f space d i a g o n a l space m a t r i x space g i v e s space t h e space m u l t i p l i c a t i o n space o f space d i a g o n a l space e l e m e n t s
D to the power of n space end exponent equals space D. D. D. D. D......... D left parenthesis U p t o space n space t i m e s right parenthesis
open square brackets table row cell d 1 end cell 0 cell..... end cell 0 row 0 cell d 2 end cell cell..... end cell cell... end cell row.. cell... end cell cell d n end cell end table close square brackets open square brackets table row cell d 1 end cell 0 cell..... end cell 0 row 0 cell d 2 end cell cell..... end cell cell... end cell row.. cell... end cell cell d n end cell end table close square brackets space.............
equals open square brackets table row cell d 1 to the power of n end cell 0 cell..... end cell 0 row 0 cell d 2 to the power of n end cell cell..... end cell cell... end cell row.. cell... end cell cell d n to the power of n end cell end table close square brackets
equals d i a g left square bracket space d 1 to the power of n space d 2 to the power of n space d 3 to the power of n space........ space space d n to the power of n space right square bracket

    Therefore the correct option is choice 3

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