Consider an n x n matrix. Suppose i wish to find the eigen values of this matrix.
Now, we know that row transformation is equivalent to a change of basis in the vector space. But, we also know that a change of basis preserves the eigen value of a matrix.
This means, if i am able to change the given matrix into an upper triangular matrix using row/column transformations, the eigen values should remain intact under change of basis.
This means, the eigen values should be found on the diagonal of the triangular matrix obtained , which is not the case. Where could the method go wrong?