3 linear equations, for what value lambda no solutions etc.

[Gregg]

Homework Statement

$$\lambda x_1 +x_2=1$$
$$x_1 +2\lambda x_2+x_3=2$$
$$x_2 +\lambda x_3=1$$

The Attempt at a Solution

The determinant is 0 for Lambda=0 and 1, found the infinite solutions associated with those to be

(1,0,1)+t(-1,1,-1)

and

(0,1,2)+t(1,0,-1)

I need to show a contradiction for some Lambda i.e. row reduction and show that 1=0 if Lambda is equal to some value. I havn't been able to do this? If it is not possible, do I need to show that the nullity is 0 for all Lambda not equal to 1 or 0 (i.e. when the determinant is 0)

 

[Mark44]

Let's look at this system of equations as a matrix product.
$$\begin{bmatrix} \lambda & 1 & 0 \\ 1 & 2\lambda & 1 \\ 0 & 1 & \lambda\end{bmatrix}\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix} 1\\2\\1\end{bmatrix}$$

In symbols, this is Ax = b.

If |A| = 0, A is not invertible, there will be either an infinite number of solutions or none at all.

In calculating |A| I get three values of $\lambda$, not two as you show.

 

[Gregg]

So |A|=0 means infinite solutions or none at all. Just check case by case? I found eigenvalues 0,1,-1.

$$\lambda = 0 \Rightarrow \text{infinitely many solutions}$$
$$\lambda = -1 \Rightarrow \text{no solutions}$$
$$\lambda = 1 \Rightarrow \text{infinitely many solutions}$$

 

[Mark44]

Yep.