Suppose the \(s \times n\) matrix on field \(K\) \[ A = {\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & & \vdots\\ a_{s1} & a_{s2} & \cdots & a_{sn}\\ \end{pmatrix}} \] satisfies \(s \le n\), and \[2|a_{ii}|>\sum_{j=1}^{n}{|a_{ij}|},\ \ (i = 1, 2, \cdots, s),\] proof: rank of \(A\)’s row vector group, \(\gamma_{1}\), \(\gamma_{2}\), ⋯, \(\gamma_{s}\), equals to \(s\).
Fibonacci sequence is a sequence as
\[1, 2, 3, 5, 8, 13, 21, 34, \cdots,\]
it satisfies: \(F_n = F_{n - 1} + F_{n - 2}(n \ge 3), F_1 = 1, F_2 = 2\)
Proof the Fibonacci numbers \(F_n\) can be get from the determinant \[ F_n = {\begin{vmatrix} 1 & -1 & 0 & 0 & \cdots & 0 & 0 & 0\\ 1 & 1 & -1 & 0 & \cdots & 0 & 0 & 0\\ 0 & 1 & 1 & -1 & \cdots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \cdots & 1 & 1 & -1\\ 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 1\\ \end{vmatrix} }; \]
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.
Consider the symmetries of \(a_{1i}\), \(a_{2j}\) 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.
Does \(f(x,y,z)=x^3+y^3+z^3-3xyz\) has a factor which is of degree 1? if yes, show it.