Determinant of $ 3\times 3$ matrix by using gauss

Question

I am trying to calculate the determinant of the following matrix by performing Gaussian elimination. I know that the determinant is $1$ but I get the wrong result.

\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 2 & 3 & -4 \end{bmatrix}

$r_2-r_1$ gives:

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & -1 \\ 2 & 3 & -4 \end{bmatrix}

$r_3-2r_1$ gives:

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & -1 \\ 0 & 1 & -6 \end{bmatrix}

Swap $r_2$ and $r_3$

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -6 \\ 0 & 0 & -1 \end{bmatrix}

Determinant$= 1.1.(-1) = -1$ which is wrong.

Answer 1

As you swap rows $r_2$ and $r_3$, you introduce a negative one in the determinant, hence the determinent is $-(-1)=1$.

Answer 2

We have $ \det \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -6 \\ 0 & 0 & -1 \end{bmatrix}= - \det \begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & -1 \\ 0 & 1 & -6 \end{bmatrix}$,

since you swapped $r_2 $ and $r_3$.