若线性方程组x1+x2=a1,x2+x3=a2,x3+x4=a3,x4+x1=a4有解,则常数a1,a2,a3,a4应满
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若线性方程组x1+x2=a1,x2+x3=a2,x3+x4=a3,x4+x1=a4有解,则常数a1,a2,a3,a4应满足关系式?
x1+x2 =a1
x2+x3 =a2
x3+x4=a3
x1 +x4=a4
增广矩阵(A,b)=
1 1 0 0 a1
0 1 1 0 a2
0 0 1 1 a3
1 0 0 1 a4
r4-r1得
1 1 0 0 a1
0 1 1 0 a2
0 0 1 1 a3
0 -1 0 1 a4-a1
r1+r4,r2+r4
1 0 0 1 a4
0 0 1 1 a2+a4-a1
0 0 1 1 a3
0 -1 0 1 a4-a1
r2r4
1 0 0 1 a4
0 -1 0 1 a4-a1
0 0 1 1 a3
0 0 1 1 a2+a4-a1
r4-r3
1 0 0 1 a4
0 -1 0 1 a4-a1
0 0 1 1 a3
0 0 0 0 a2+a4-a1-a3
因为线性方程有解
所以a2+a4-a1-a3=0
答案:a2+a4-a1-a3=0
x2+x3 =a2
x3+x4=a3
x1 +x4=a4
增广矩阵(A,b)=
1 1 0 0 a1
0 1 1 0 a2
0 0 1 1 a3
1 0 0 1 a4
r4-r1得
1 1 0 0 a1
0 1 1 0 a2
0 0 1 1 a3
0 -1 0 1 a4-a1
r1+r4,r2+r4
1 0 0 1 a4
0 0 1 1 a2+a4-a1
0 0 1 1 a3
0 -1 0 1 a4-a1
r2r4
1 0 0 1 a4
0 -1 0 1 a4-a1
0 0 1 1 a3
0 0 1 1 a2+a4-a1
r4-r3
1 0 0 1 a4
0 -1 0 1 a4-a1
0 0 1 1 a3
0 0 0 0 a2+a4-a1-a3
因为线性方程有解
所以a2+a4-a1-a3=0
答案:a2+a4-a1-a3=0
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