Exercises of determinant

  1. 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.

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

    \[ x^3+y^3+z^3-3xyz = {\begin{vmatrix} x & y & z\\ y & z & x\\ z & x & y \end{vmatrix}} \] 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.