determinant-1-0-0-0-1-1-1-0-0-0-0-1-1-0-0-0-0-1-1-0-0-0-0-1-1-1-0-0-0-1-

Question Number 218119 by MrGaster last updated on 30/Mar/25

| \begin{matrix} ε & 1 & 0 & 0 & 0 & 1 \\ 1 & ε & 1 & 0 & 0 & 0 \\ 0 & 1 & ε & 1 & 0 & 0 \\ 0 & 0 & 1 & ε & 1 & 0 \\ 0 & 0 & 0 & 1 & ε & 1 \\ 1 & 0 & 0 & 0 & 1 & ε \end{matrix} | = ?

Answered by SdC355 last updated on 30/Mar/25

Holy shit \dots . what is that Lol

But that \det {A} is ε^{6} - 6 ε^{4} + 9 ε^{2} - 4 \dots

Commented by MrGaster last updated on 30/Mar/25

THANK!

Commented by Ghisom last updated on 30/Mar/25

the “ holy {shit}^{″} is back . nice .

Answered by Wuji last updated on 30/Mar/25

f o r t h e 6 \times 6 m a t r i x

M (ε) = ε I + A

f o r a c i r c u l a n t g r a p h C_{n} w e h a v e e i g e n v a l u e s

o f A :

2 c o s (\frac{2 π k}{n}) \Rightarrow k = 0, 1, \dots, n - 1

f o r n = 6

2, 1, - 1, - 2, - 1, 1

M (ε) = ε I + A

ε + 2, {(ε + 1)}^{2}, {(ε - 1)}^{2}, (ε - 2)

d e t M (ε) = (ε + 2) {(ε + 1)}^{2} {(ε - 1)}^{2} (ε - 2)

d e t M (ε) = (ε^{2} - 4) {(ε^{2} - 1)}^{2}

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