Question

问a,b各取何值时,线性方程组无解,有唯一解,有无穷多解?在有无穷多解时求出其通解.

Answer 1

增广矩阵 (A, β) =

[ 1 1 1 3 0]

[ 2 1 3 5 1]

[ 3 2 a 7 1]

[ 1 -1 3 -1 b]

行初等变换为

[ 1 1 1 3 0]

[ 0 -1 1 -1 1]

[ 0 -1 a-3 -2 1]

[ 0 -2 2 -4 b]

行初等变换为

[ 1 1 1 3 0]

[ 0 1 -1 1 -1]

[ 0 0 a-4 -1 0]

[ 0 0 0 -2 b-2]

当 a≠4 时,|A|=2(4-a)≠0, 方程组有唯一解.

当 a=4 时,|进一步行初等变换为

[ 1 1 1 3 0]

[ 0 1 -1 1 -1]

[ 0 0 0 1 0]

[ 0 0 0 0 b-2]

当 a=4,b≠2 时,r(A)=3, r(A, β) =4, 方程组无解.

当 a=4,b=2 时,r(A)=r(A, β) =3