Do elementary transformations affect the eigenvalues?

Question

I have a general doubt regarding the eigenvalues of a diagonal matrix. I know that the eigenvalues of a diagonal matrix are the diagonal entries themselves, but do elementary transformations affect the eigenvalues?

We know that every matrix can be reduced to a diagonal matrix via elementary transformations. The matrix we obtain after applying the elementary Transformations will have the same eigenvalues as the original matrix that we started with.

Answer 1

Eigen values change when you apply elementary transformation.

Example $$\begin{bmatrix}2 & 1\\0&1\end{bmatrix}$$ This has eigen values 2 and 1 now do elementary tranformation $R_1\rightarrow0.5R_1-0.5R_2$, The matrix becomes $$\begin{bmatrix}1 & 0\\0&1\end{bmatrix}$$ with eigenvalues 1, 1