I found the following proof that the trace of a square matrix is equal to the sum of its eigenvalues (see the second half of page 3):
https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf
The first half of page 3 contains a proof that show that the determinant of a square matrix is equal to the product of its eigenvalues; I understand all of this.
I am having severe difficulty understanding the second half of page 3, which is the proof that the trace of a square matrix is equal to the sum of its eigenvalues.
My immediate difficulty is in understanding the following section:
In order to get the $\lambda^{n−1}$ term, the $\lambda$ must be chosen from $n − 1$ of the factors, and the constant from the other. Hence, the $\lambda^{n−1}$ term will be $-\lambda_1\lambda^{n - 1} - ... - \lambda \lambda^{n - 1} = -(\lambda_1 + ... + \lambda_n)\lambda^{n−1}$. Thus $c_{n−1} = −(\lambda_1 +···+ \lambda_n)$.
I would greatly appreciate it if people could please clarify what this is saying. What is meant by "In order to get the $\lambda^{n−1}$ term, the $\lambda$ must be chosen from $n − 1$ of the factors, and the constant from the other."? What is the entire section trying to say?