On 1 April 2011 I wrote to Wolfram Research:

I had a look at Wolfram Alpha and was slightly surprised that [it] returned, among many useful information about the matrix

12

01the assertion that it had eigenvectors

(1,0) and (0,0).

You and I, of course, understand how to interpret the answer, but it could be quite confusing for a first year student.

I tried it again today, I got the same answer; see for yourself.

Update 12 February 2015: Wolfram Alpha still returns the zero vector as an eigenvector for a matrix.