Question

线性代数,当t为何值时,线性方程组 有无穷多解,并求出此线性方程组的通解.

Answer 1

|A| =

|1 1 t|

|1 -1 2|

|-1 t 1|

|A| =

|1 2 t-2|

|1 0 0|

|-1 t-1 3|

|A| = (-1)*

| 2 t-2|

|t-1 3|

A| = -[6-(t-1)(t-2)]=0,得 t=4,-1.

当 t=-1 时,(A,b) =

[1 1 -1 4]

[1 -1 2 -4]

[-1 -1 1 1]

行初等变换为

[1 1 -1 4]

[0 -2 3 -8]

[0 0 0 5]

r(A)=2,r(A,b)=3,方程组无解.

当 t=4 时,(A,b) =

[1 1 4 4]

[1 -1 2 -4]

[-1 4 1 16]

行初等变换为

[1 1 4 4]

[0 -2 -2 -8]

[0 5 5 20]

行初等变换为

[1 0 3 0]

[0 1 1 4]

[0 0 0 0]

r(A)=r(A,b)=2