the eigen value for upper triangular/lower triangular/diagonal matrices are the diagonal elements of the matrix.
it is true.
I know for determinant, applying row transformation or column transformation value of the determinant remains the same.
if R_{i} ---> k.R_{i} , then determinant is changed.
if R_{i} ---> R_{i} + (k.R_{j}) , then Determinant is not changed.
Note that, if A and B , Determinants is equal, then Eigen values are need not to be same for A and B.
$\begin{pmatrix} 2 & -1\\ -4 & 5 \end{pmatrix}$ ---------> (i)
after applying row transformation R_{1} ---> R_{1} + ($\frac{1}{5}$R_{2})
$\begin{pmatrix} \frac{6}{5} & 0\\ -4 & 5 \end{pmatrix}$ ---------> (ii)
(i) and (ii) have determinant value same(=6),
but eigen values of (i) is 1,6 and eigen values of (ii) is $\frac{6}{5} $,5 which are different.