A square matrix A is said to be nilpotent of index m. $I f A^{m} = 0, n o w, i f f o r t h i s A, (I - A)^{n} = I + A + A^{2} + . . . + A^{m - 1}, t h e n n i s e q u a l t o$

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Solution

The correct option is D - 1
$L e t B = I + A + A^{2} + . . . + A^{m - 1}$
$\Rightarrow B (I - A) = (I + A + A^{2} + . . . A^{m - 1}) (I - A)$
$= I - A^{m} = I$
$\Rightarrow B = (I - A)^{- 1} \Rightarrow n = - 1$

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I f A^{m} = 0, n o w, i f f o r t h i s A, (I - A)^{n} = I + A + A^{2} + . . . + A^{m - 1}, t h e n n i s e q u a l t o

A^{m} = 0

(I - A)^{n} = I + A + A^{2} + A^{3} + . . . . + A^{m - 1}

I f A^{m} = 0, n o w, i f f o r t h i s A, (I - A)^{n} = I + A + A^{2} + . . . + A^{m - 1}, t h e n n i s e q u a l t o

I f A^{m} = 0, n o w, i f f o r t h i s A, (I - A)^{n} = I + A + A^{2} + . . . + A^{m - 1}, t h e n n i s e q u a l t o

A square matrix A is called Nilpotent if there exists a positive integer m such that $A^{m}$ = I.

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