Question

show that a 2x2 complex matrix A is nilpotent if and only if Tr(A)=0 and Tr(A^2)=0. give an example of a complex 2x2 matrix which is not nilpotent but whose trace is 0

Answer 1

In general the trace of a matrix is the sum of the eigenvalues in the algebraic closure. Suppose λ is an eigenvalue of A and let v≠0 be such that
Av=λv
Suppose An=0 for some n. Then
Anv=λnv=0
Since v≠0, we have that λn=0, hence λ=0. Thus all eigenvalues of A are zero hence the trace is zero.

Let A be an n×n complex nilpotent matrix. Then we know that because all eigenvalues of A must be 0, it follows that tr(An)=0 for all positive integers n.

If the eigenvalues of A are λ1, …, λn, then the eigenvalues of A^k are λ1^k, …, λn^k. It follows that if all powers of A have zero trace, then

for all k≥1.
Using Newton's identities to express the elementary symmetric functions of the λi's in terms of their power sums, we see that all the coefficients of the characteristic polynomial of A (except that of greatest degree, of course) are zero. This means that A is nilpotent.