Row swapping not working as expected for determinant

Question

Suppose I have determinant A such that $$ A= \begin {vmatrix} 1&0&1 \\ 5&-1&0 \\ 1&0&0 \end {vmatrix} $$

I conduct row operation $ R_1 <-> R_3 $ So I should have $$ A = - \begin {vmatrix} 1&0&0 \\ 5&-1&0 \\ 1&0&1 \end {vmatrix} $$

but that does not seem to be the case because in the first case A=-1 and in the other A=1
That isn't working as expected. What am I missing here?

Answer 1

It is$$\begin {vmatrix} 1&0&1 \\ 5&-1&0 \\ 1&0&0 \end {vmatrix}=1$$ and

$$\begin {vmatrix} 1&0&0 \\ 5&-1&0 \\ 1&0&1 \end {vmatrix}=-1.$$ So

$$\begin {vmatrix} 1&0&1 \\ 5&-1&0 \\ 1&0&0 \end {vmatrix}=-\begin {vmatrix} 1&0&0 \\ 5&-1&0 \\ 1&0&1 \end {vmatrix}.$$