线性代数解方程组解下列非其次线性方程组：20X1+10X2+10X3+15X4=705X1+5X2+10X3+15X4=355X1+15X2+5X3+10X4=358X1+10X2+10X3+20X4=50

Question

线性代数解方程组
解下列非其次线性方程组：
20X1+10X2+10X3+15X4=70
5X1+5X2+10X3+15X4=35
5X1+15X2+5X3+10X4=35
8X1+10X2+10X3+20X4=50

Answer 1

最好用矩阵解.
20X1+10X2+10X3+15X4=70 (1)
5X1+5X2+10X3+15X4=35 (2)
5X1+15X2+5X3+10X4=35 (3)
8X1+10X2+10X3+20X4=50 (4)
(1)-(4)*2.5,(2)-(3),(3)*4-(1) 得
0 X1 -15 X2 -15 X3 -35 X4= -55 (5)
0 X1 -10 X2+ 5 X3+ 5 X4= 0 (6)
0 X1+ 50 X2+ 10 X3+ 25 X4= 70 (7)
8 X1+ 10 X2+ 10 X3+ 20 X4= 50 (4)
(5)*2-(6)*3,(6)*5+(7) 得
0 X1+ 0 X2 -45 X3 -85 X4= -110 (8)
0 X1+ 0 X2+ 35 X3+ 50 X4= 70 (9)
0 X1+ 50 X2+ 10 X3+ 25 X4= 70 (7)
8 X1+ 10 X2+ 10 X3+ 20 X4= 50 (4)
(8)*7+(9)*9 得
0 X1+ 0 X2+ 0 X3 -145 X4= -140 (10)
0 X1+ 0 X2+ 35 X3+ 50 X4= 70 (9)
0 X1+ 50 X2+ 10 X3+ 25 X4= 70 (7)
8 X1+ 10 X2+ 10 X3+ 20 X4= 50 (4)
由(10)得 X4=28/29
代入(9)得 X3=18/29
代入(7)得 X2=23/29
代入(4)得 X1=60/29
实际就是用加减消元法,化为阶梯形.
解法2:
用EXCEL的矩阵函数解.
输入矩阵A:
20 10 10 15
5 5 10 15
5 15 5 10
8 10 10 20
用MINVERSE 函数得出A的逆阵A-:
0.06897 -0.06897 -0.03448 0.01724
-0.00690 0.00690 0.10345 -0.05172
0.02069 0.37931 0.08966 -0.34483
-0.03448 -0.16552 -0.08276 0.24138
输入矩阵B:
70
35
35
50
用MMULT函数计算A-与B的乘积:
2.0689655 ...X1
0.7931034 ...X2
0.6206897 ...X3
0.9655172 ...X4
就是方程组的解