Note that matrix A is said to be Nilpotent if
where m is any integer and
is a null matrix of same order as of A.
Lets take example of matrix A which is nilpotent.
Therefore, we can see that
, Hence, the matrix A is nilpotent. Similarly, we can take other examples of Nilpotent matrices. Note that we may or may not have m=2 such that
but we can also have
such that
.
saakshi says
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saakshi says
loved it
Jashan says
Thanks @saakshi
Dirgharaj Karki says
But I can’t understand that,if 1 is a positive integer than why A^1=0 is not possible on your example?Does it reveal the definition?Please waiting for your answer.
Jashan says
Hello,
By Nilpotent matrix, we mean any matrix A such that A^m = 0 where m can be any specific integer. It does not mean that A^m=0 for every integer.
A^m=0 may be true for just m=3 but not for m=1 or m=2. The matrix A would still be called Nilpotent Matrix.
If, you still have problem in understanding then please feel free to write back.
Syed Danish says
SIR G….PLZ HELP…
If A is a nilpotent matrix of order 3 with index of nilpotency 2 & trace of a = 3 then trace of 2A7+3A5+4A2+A+I=………
…
2A7…MEANS 2 RAISE TO POWER 7….SO ON….
Raymond says
Like it