What is the determinant of matrix?

Question

Find determinant of the $n \times n$ permutation matrix $$ M= \left[ {\begin{array}{cccc} 0 & 0 & \ldots & 0 & 1\\ 0 & 0 & \ldots & 1 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & 0 & \ldots & 0 & 0\\ \end{array} } \right] $$

My answer was $(-1)^n$. Am I correct

Answer 1

Remember cofactor expansion: $$\det \left[ {\begin{matrix} {\color{red}0} & {\color{red}0} & \ldots & {\color{red}0} & {\color{red}1}\\ 0 & 0 & \ldots & 1 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 1 & \ldots & 0 & 0\\ 1 & 0 & \ldots & 0 & 0\\ \end{matrix} } \right] = $$

$${\color{red}0}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 1\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] - {\color{red}0}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 1\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] + {\color{red}0}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 1\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] +\cdots $$

$$\cdots + (-1)^{n-2}{\color{red}0}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 0\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] + (-1)^{n-1}{\color{red}1}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 1\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] $$

So the only thing that survives is

$$(-1)^{n-1}\cdot \det \left[ {\begin{matrix} 0 & \ldots & 0 & 1\\ \vdots & \ddots & \vdots & \vdots\\ 0 & \ldots & 0 & 0\\ 1 & \ldots & 0 & 0\\ \end{matrix} } \right] $$

Performing this $n$ times you will get $(-1)^{n(n-1)/2}$

Answer 2

i think the answer is $det\ \pmatrix{0 & 0 & \cdots & 0 & 1\\0 & 0& \cdots & 1 & 0\\ \vdots& \vdots& \ddots&\vdots&\vdots\\1&0&\cdots&0&0} = \left\{ \begin{array}{ll} (-1)^{(n-1)/2} & \mbox{ if $n$ is odd }\\ (-1)^{n/2} & \mbox{ if $n$ is even} \end{array} \right.$

here is how you see this. let $\lambda$ be an eigenvalue of $M.$ then you have $$\{x_n = \lambda x_1, x_1 = \lambda x_n\}, \{x_{n-1} = \lambda x_2, x_2 = \lambda x_{n-1}\}, \cdots $$ these come in pairs except when $n$ is odd you are left with single equation for the middle $x_{(n+1)/2}$. for each pair you get the the eigenvalues $\pm 1$ and $1$ for the single one. now the determinant of $M$ is the product of $-1$ for each pair, hence the formula.