Matrix determinant operations.

Suppose you are trying to find the determinant of the following matrix using the "upper triangulation" method:

$\begin{matrix} 1&0&0\\ 0&1&0\\ 1&1&1 \end{matrix}$

If I take R1 - R3 -> R3 (row 1 minus row 3 and put the result in row 3):

$\begin{matrix} 1&0&0\\ 0&1&0\\ 0&-1&-1 \end{matrix}$

Then I do R2 + R3 ->R3:

$\begin{matrix} 1&0&0\\ 0&1&0\\ 0&0&-1 \end{matrix}$

And the determinant is then = -1 (because I multiply the elements in the diagonal)

However if the first step is instead:

R3 - R1 ->R3:

$\begin{matrix} 1&0&0\\ 0&1&0\\ 0&1&1 \end{matrix}$

And then R3 - R2 -> R3:

$\begin{matrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{matrix}$

The determinant is 1 !!!

How do you know which row is subtracted from what?

• Somewhere you interchanged two rows, which means you change the orientation so then the determinant changed sign. – Kaladin Mar 8 '14 at 21:19
• your mistake is (if i understand you correctly): if $range\left(A\right)$ = $range\left(B\right)$ that doesn't mean that $det\left(A\right)=det\left(B\right)$. (the line operations you apply are keeping the range constant) – Max Mar 8 '14 at 21:20