设有3*4非齐次线性方程组AX=β,若 ？ 成立,则该方程组一定有解。大一线性代数问题求解啊

设有3*4非齐次线性方程组AX=β,若 ？ 成立,则该方程组一定有解。大一线性代数问题求解啊

r(A)=r（A，β），就是系数矩阵的秩等于增广矩阵的秩

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