在齐次线性方程组中,为什么系数行列式D=0,奇次线性方程组才有非零解

2025-03-16 08:53:33
推荐回答(2个)
回答1:

首先必须区几概念:线性程组、齐程组非齐程组
线性程组总称凡写形式程组都统称线性程组
a11*X1 + a12*X2 + …… + a1n*Xn = b1
a21*X1 + a22*X2 + …… + a2n*Xn = b2
………………
am1*X1 + am2*X2 + …… + amn*Xn = bm

线性程组齐程组非齐程组两种
1. 数项b1、b2、……、bm全零该程组称齐程组
2. 数项b1、b2、……、bm全零该程组称非齐程组

另外系数行列式够准确行数m(程数)与列数n(未知元数)相等系数矩阵才能取行列式计算般用系数矩阵讨论更准确考虑矩阵秩

*********************于线性程组性质*********************
(设D系数矩阵b数项向量r(D)表示矩阵D秩r(D,b)表示增广矩阵(D,b)秩)
1. r(D)=r(D,b)<列秩n 构系数矩阵列向量组线性相关则线性程组数解;
2. r(D)=r(D,b)=列秩n 构系数矩阵列向量组线性关则线性程组存唯解;
3. r(D) ≠ r(D,b) 线性程组解

*****************关于矩阵秩行列式值否零关系*******************
(设|D|表示矩阵D行列式)
特别 系数矩阵 行数m=列数n存r(D) ≠ r(D,b) 情况
1. |D| = 0或者r(D)=r(D,b)<列秩n 系数向量组线性相关则线性程组数解;
2. |D| ≠ 0或者r(D)=r(D,b)=列秩n 系数向量组线性关则线性程组存唯解

********************于齐程组********************
齐程组看作线性程组种特殊形式即数向量b零向量特殊情况
同存r(D) ≠ r(D,b) 情况(假设m=n)

1. |D| = 0或者r(D)=r(D,b)<列秩n 系数向量组线性相关则齐程组非零解(即除零解外数非零解);
2. |D| ≠ 0或者r(D)=r(D,b)=列秩n 系数向量组线性关则线性程组存唯解解零解

面我章致理解明白给我留言我再补充~~

回答2:

D=0 至少有一行或列是0 少一个方程无解 少一个未知数 无穷多解

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