Determinant

Suppose \(n \ge 2\)，show that if the elements of a \(n \times n\) matrix \(A\) are \(1\) or \(-1\), then \(|A|\) must be an even number.

\(Proof.\) Consider the symmetries of \(a_{1i}, a_{2j} \text{ and } a_{1j}, a_{2i}\), the value of corresponding determinant of the \(2 \times 2\) matrix must be an even number, so the n-order determinant must be an even number. \(\blacksquare\)

Does \(f(x,y,z)=x^3+y^3+z^3-3xyz\) has a factor which is of degree 1? if yes, show it.

\(Solution.\) \[ x^3+y^3+z^3-3xyz = {\begin{vmatrix} x & y & z\\ y & z & x\\ z & x & y \end{vmatrix}} \] this determinant can be transformed to \[ (x+y+z) \times {\begin{vmatrix} 1 & 1 & 1\\ y & z & x\\ z & x & y \end{vmatrix}} \] so \((x+y+z)\) is the factor of degree \(1\). \(\blacksquare\)