Trace of a nilpotent matrix is zero.
121987
Direct proof that nilpotent matrix has zero trace
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
78298
On the nilpotence of the matrix $AB-BA$
133758
Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.
88473
Proof that the trace of a matrix is the sum of its eigenvalues
98646
Show determinant of matrix is non-zero
90506
Finite multiplicative group of matrices with sum of traces equal to zero.
101631
Determinant of a finite-dimensional matrix in terms of trace
35801
Traceless matrices and commutators
75002
show that $\det(A)=0$ in this case
30037
What information does the trace of a matrix give?
18036
Localization by a non-nilpotent as a polynomial quotient
128043
$A^n=0$ for $A$ nilpotent without the use of minimal polynomial
111339
Why the determinant of even-sized magic matrices are zero? Anything special?
145161
Testing whether an element of a tensor product of modules is zero
105141
$\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$
51800
We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$