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I'm a bit confused about the following: According to the rules of determinants, interchanging two rows should affect the determinant by a minus sign. Now, take the identity matrix $I_3$, if we interchange the first row by the third row, you'd have $$A = \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{pmatrix}$$

Would this mean that the determinant is being affected? We are interchanging two rows, so the determinant should be $-\det(I_3)$, yet according to my lecture slides, it's $(-1)^{3-1} \det(I_3) = 1\det(I_3)$ is this a mistake or am I not understanding something?

Thanks in advance!

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    $\begingroup$ the slides are wrong $\endgroup$
    – janmarqz
    Jun 17, 2021 at 3:53
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    $\begingroup$ What you’ve written from your lecture slides makes no sense. You can see easily enough directly that the det is $-1$. $\endgroup$
    – Ted Shifrin
    Jun 17, 2021 at 3:54
  • $\begingroup$ I will let them know, thanks guys $\endgroup$
    – JakeDrone
    Jun 17, 2021 at 3:55
  • $\begingroup$ @JakeDrone Can you post the actual slide? $\endgroup$
    – littleO
    Jun 17, 2021 at 4:04
  • $\begingroup$ @littleO unfortunately not, I don't think the uni allows it. It's not really a slide, on our math courses we have exercises made for us, we answer them and in turn get an explanation for each exercise. In this case, the exercise gave me the above explanation, although its rare, mistakes happen (these are usually done by TA's). $\endgroup$
    – JakeDrone
    Jun 17, 2021 at 4:05

1 Answer 1

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The lecture slides are incorrect, the determinant is -1.

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