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Prove that the trace of a nilpotent matrix is always zero and that the trace of any power of that matrix is also zero.

Prove that the trace of a nilpotent matrix is always zero and that the trace of any power of that matrix is also zero.

Grade:12

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
LetAbe anilpotent matrix. AssumeAn=0. Letλbe aneigenvalueofA. ThenAx=λxfor some nonzerovectorx. Byinductionλnx=Anx=0, soλ=0.
Conversely, suppose that all eigenvalues ofAarezero. Then the chararacteristicpolynomialofA:det(λI−A)=λn. It now follows from the Cayley-Hamilton theorem thatAn=0.
Since thedeterminantis theproductof the eigenvalues it follows that a nilpotent matrix has determinant 0. Similarly, since thetraceof a square matrix is thesumof the eigenvalues, it follows that it has trace 0.
As in nilpotent matrix An=0 it also follows that trace(An)=0 similarly

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