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?$