does a row operation preserve the eigenvalues or eigenvectors of a matrix?

Question

justify or give counterexamples as appropriate.

Answer 1

Row operations do not preserve eigenvalues. In a diagonal matrix the eigenvalues are the entries along the diagonal. If you perform a scaling operation, these numbers change thus changing the eigenvalues.

Answer 2

Of course not. Consider the matrix
[1 2]
[2 5]. The eigenvalues are 3 +- 2*sqrt(2). Now add -2 times the first row to the second:

[1 2]
[0 1]. The eigenvalues are 1.

Answer 3

Row Operation Matrix

Answer 4

Thanks you have just answered my question as well, row reducing is definately not permitted in finding eigenvalues