I have the matrix
$ A = \begin{pmatrix} 1 & 2 & 3 & 4\\ -1 & 1 & 2 & 3\\ 1 & -1 & 1 & 2\\ -1 & 1 & -1 & 1\\ \end{pmatrix} $
I should be able to add multiples of any row to another row and not affect the determinant.
By adding $1$ times the first row to the second row, $-1$ times the first to the third, and $1$ times the first to the fourth, I should still have the same determinant.
This results in the matrix
$ B = \begin{pmatrix} 1 & 2 & 3 & 4\\ 0 & 3 & 5 & 7\\ 0 & -3 & -2 & -2\\ 0 & 3 & 2 & 5\\ \end{pmatrix} $
However,
$\det(A) = 27,\quad \det(B) = 87$
What am I missing? I am expecting $\det(B) = 27$ since I am only adding multiples of the first row to the others.