I want to ask if adding the rows would somehow affect my eigenvalues , I just added 1st row to 3rd row and after finding the eigenvalues for both $A$ and $A'$ I didn't get the same result.

Adding the rows doesn't not affect the determinant nor the rank, so why should eigenvalues be different ?

$A=$$\begin{bmatrix}1&2&1\\6&-1&0\\-1&-2&-1 \end{bmatrix}$

$A'=$$\begin{bmatrix}1&2&1\\6&-1&0\\0&0&0 \end{bmatrix}$

Eigenvalues of A' : $0,\sqrt{13},-\sqrt{13}$

Eigenvalues of A : $0,3,-4$