Question

does performing row operations on matrix A change the eigenvalues and eigenspace of (A-lamba*I)?

Answer 1

yes. For example, take the 2x2 identity matrix

1 0
0 1

It clearly has eigenvalues 1 and the eigenspace is R^2

But after two row operations, adding R1 to R2 and adding the resulting R2 back to R1, we have

2 1
1 1

The characteric polynominal of this matrix is
\[(2 - \lambda)(1 - \lambda) - 1 = \lambda^2 - 3\lambda + 1\]
which has completely different eigenvalues and eigensapces.