Math Help - find this eigenvalue

1. find this eigenvalue

Find corresponding eigenvalue

given: A= 2 -1 -1 x=1
-1 2 -1 1
-1 -1 2 1

I figured that this matrix multiplied with that vector results in zero, hence eigenvalue can be zero, but I also figured that eigenvalue can be 3. Im now confused about possible eigenvalue. Help plz

2. a matrix can (and usually does) have more than one eigenvalue.

in this case, all you need to do is find the particular eigenvalue for a particular eigenvector, which you have done.

3. If <1, 1, 1> is, in fact, an eigenvector for this matrix, then, by definition of "eigenvector", we must have
[2 -1 -1][1] [1]
[-1 2 -1][1]= L[1]
[-1 -1 2][1] [1]

Multiply the left side and see if it does give a multiple of <1, 1, 1> and what that multiple is.

You did not need to calculate the possible eigenvalues in advance to do that.