As far from textbook I heard about this formula for calculating eigenvalues of a square matrix:
- det(M)=$\lambda_1\lambda_2\lambda_3 ....$
- Tr(M)=$\sum_i \lambda_i $
Ok I have no doubt about the determinant formula. But I have some confusion about the trace formula as it does not hold always. Whenever the trace changes due to matrix operations I can not decide which trace to use and which I should not. Take this example-
Normally the trace formula is working here. Ok fine. But if I do some matrix operation (like $C_3=C_3-C_2$), the trace changes (from $12$ to $13$). And I can not use the formula anymore.
My question is how do I know if the formula will work or not?