Determinant - Applications

Circulants

Second order

$\left| \begin{array}{cc} x_1 & x_2 \\ x_2 & x_1 \end{array} \right|=\left(x_1+x_2\right)\left(x_1-x_2\right).$

Third order

$\left| \begin{array}{ccc} x_1 & x_2 & x_3 \\ x_3 & x_1 & x_2 \\ x_2 & x_3 & x_1 \end{array} \right|=\left(x_1+x_2+x_3\right)\left(x_1+\omega x_2+\omega ^2x_3\right)\left(x_1+\omega ^2x_2+\omega x_3\right),$

where ω and ω2 are the complex cube roots of 1. In general, the nth-order circulant determinant is

$\left| \begin{array}{ccccc} x_1 & x_2 & x_3 & \cdots & x_n \\ x_n & x_1 & x_2 & \cdots & x_{n-1} \\ x_{n-1} & x_n & x_1 & \cdots & x_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_2 & x_3 & x_4 & \cdots & x_1 \end{array} \right|=\prod _{j=1}^n \left(x_1+x_2\omega _j+x_3\omega _j^2+\ldots +x_n\omega _j^{n-1}\right),$

where ω_j is an nth root of 1.

Read more about this topic: Determinant, Applications

Determinant - Applications - Circulants