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

True

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False

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Solution

The correct option is B
False

We have defined a nilpotent matrix in the video. A square matrix ‘A’ is called nilpotent if there exists an integer m such that A^m = O where O is the null matrix of the same order as of A.

$A^{m}$ = I is not a condition for nilpotent matrix.

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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}

A square matrix A such that $A^{k} = A$ for a positive integer k is called a periodic matrix.

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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