1
$\begingroup$

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?

$\endgroup$
  • 4
    $\begingroup$ You multiplied R2 by -1 in your second step. $\endgroup$ – Ian Aug 22 '16 at 3:21
2
$\begingroup$

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

means $$R_j \leftarrow R_j +cRi$$

It does not include the case when

$$R_j \leftarrow R_i + cR_j$$

$\endgroup$
  • $\begingroup$ THIS is the piece of information i was missing. thank you. $\endgroup$ – Matt Creighton Aug 22 '16 at 3:28
0
$\begingroup$

In your three steps, you are multiplying the row by $-1$ in each step, so your determinant changes by $(-1)^3=-1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.