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I have a matrix $M$ that is equal to: $\begin{bmatrix} 1 & 1 & 0\\ 1 & 0 & 1 \\ 0 & 1 & 1\\ \end{bmatrix}$

It's easy to compute $|M| = -2$, but then given matrix:

$ N = \begin{bmatrix} M & 0 \\ 0 & M \\ \end{bmatrix}$

What is $|N|$? My textbook and multiple sources say it's $ |N| = -1$. But I couldn't find any explanation as to why. Wikipedia's Determinants page says it's $det(N) = det(M)det(M)$ which is 4. That makes sense to me. Why it says -1 in the solutions manual does not. I've even tried expanding the matrix out into a 6x6 matrix and getting the determinant of that (used calculator to make sure), but it's still 4.

$ det(N) = |\begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1\\ \end{bmatrix}| = 4?$

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    $\begingroup$ You are completely right, $\det(N)=(-2)^2=4$. Did you check the matrix $M$ for typos? $\endgroup$ – Dietrich Burde Oct 16 '18 at 21:32
  • $\begingroup$ Yes. Very carefully. Thank you for reaffirming my sanity $\endgroup$ – QuantumHoneybees Oct 16 '18 at 21:53
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    $\begingroup$ Moreover, $|N|=-1$ can’t be correct since that would mean that $|M|$ is imaginary. $\endgroup$ – amd Oct 16 '18 at 22:47

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