【3阶范德蒙行列式计算】
【3阶范德蒙行列式计算】
D3=∣111x1x2x3x12x22x32∣D_3= \begin{vmatrix} 1 & 1 & 1 \\ x_1 & x_2 & x_3 \\ x_1^2 & x_2^2 & x_3^2 \end{vmatrix} D3=∣∣1x1x121x2x221x3x32∣∣
D3=r2(−x1)+r3∣111x1x2x30x2(x2−x1)x3(x3−x1)∣D_3\xlongequal{r_2(-x_1)+r_3} \begin{vmatrix} 1 & 1 & 1 \\ x_1 & x_2 & x_3 \\ 0 & x_2(x_2-x_1) & x_3(x_3-x_1) \end{vmatrix} D3r2(−x1)+r3∣∣1x101x2x2(x2−x1)1x3x3(x3−x1)∣∣
D3=r1(−x1)+r2∣1110x2−x1x3−x10x2(x2−x1)x3(x3−x1)∣D_3\xlongequal{r_1(-x_1)+r_2} \begin{vmatrix} 1 & 1 & 1 \\ 0 & x_2-x_1 & x_3-x_1 \\ 0 & x_2(x_2-x_1) & x_3(x_3-x_1) \end{vmatrix} D3r1(−x1)+r2∣∣1001x2−x1x2(x2−x1)1x3−x1x3(x3−x1)∣∣
D3=∣x2−x1x3−x1x2(x2−x1)x3(x3−x1)∣D_3= \begin{vmatrix} x_2-x_1 & x_3-x_1 \\ x_2(x_2-x_1) & x_3(x_3-x_1) \end{vmatrix} D3=∣∣x2−x1x2(x2−x1)x3−x1x3(x3−x1)∣∣
D3=(x2−x1)∣1x3−x1x2x3(x3−x1)∣D_3=(x_2-x_1) \begin{vmatrix} 1 & x_3-x_1 \\ x_2 & x_3(x_3-x_1) \end{vmatrix} D3=(x2−x1)∣∣1x2x3−x1x3(x3−x1)∣∣
D3=(x2−x1)(x3−x1)∣11x2x3∣D_3=(x_2-x_1)(x_3-x_1) \begin{vmatrix} 1 & 1 \\ x_2 & x_3 \end{vmatrix} D3=(x2−x1)(x3−x1)∣∣1x21x3∣∣
D3=(x2−x1)(x3−x1)(x3−x2)D_3=(x_2-x_1)(x_3-x_1)(x_3-x_2) D3=(x2−x1)(x3−x1)(x3−x2)
D3=∏1⩽j<i⩽3(xi−xj)D_3=\prod_{1\leqslant j<i\leqslant 3}(x_i-x_j) D3=1⩽j<i⩽3∏(xi−xj)