# basic matrix determinant properties seem inconsistent

In my maths lecture notes it gives me these rules for the determinant of a matrix:

-If two rows or columns of a matrix are interchanged, the determintant is multiplied by -1

-If a multiple of one row/column is added to another row/column, the determinant is unchanged

-If a row/column is multiplied by a real number a, the determinant is also multiplied by a

Unless theres something ive misunderstood, it seems that the second rule is inconsistent with the other two! i can swap two rows just using scale and add;

R1 <- R1 + R2

R2 <- R1 + (-1)*R2

R1 <- R1 + (-1)*R2

rule 2 says this should not affect the determinant. rule 1 says the determinant should be multiplied by -1! obviously i have missed something. Can anyone help?

• You multiplied R2 by -1 in your second step. – Ian Aug 22 '16 at 3:21

means $$R_j \leftarrow R_j +cRi$$
$$R_j \leftarrow R_i + cR_j$$
In your three steps, you are multiplying the row by $-1$ in each step, so your determinant changes by $(-1)^3=-1$.