When trying to find the determinant for a square matrix of a normal linear map such as $T: V\rightarrow W$, it's possible to just use elementary row operation to make the square matrix become an upper triangular form and time all the values on the diagonal together.
However, when the $T$ is an operator, $T: V \rightarrow V$, finding the determinant of this matrix for the operator by putting the matrix to an upper triangular form is also an idea, but just using row operation won't give a correct determinant value. Since once the basis of the codomain changes, the basis for the domain will also change. Even though I know we can also find the determinant by just using elementary row operation. The rules are if two rows are switched, then we need to time -1 to the determinant;... However, this is not the way I want to use.
The way I want to use to calculate the determinant of the operator matrix is to find the characteristic polynomial of it.
For example, the matrix is
A=$\begin{pmatrix} 2 & 6 & -15\\ 1 & 1 & -5\\ 1 & 2 & -6\\ \end{pmatrix}$
and in order to find the $P^A_{char}$:
A-$\lambda$= $\begin{pmatrix} 2-\lambda & 6 & -15\\ 1 & 1-\lambda & -5\\ 1 & 2 & -6-\lambda\\ \end{pmatrix}$
Then, at this time, can I use row operation to put this $\lambda$ matrix to become an upper triangular matrix to find the characteristic polynomial？ Since A-$\lambda$ is also an operator, I think just using row operation won't be correct, but after some trials (if there are no calculation mistakes) the row operation indeed works. Thus, I think I must misunderstand something. Any helps? Thanks in advance