You seem to be under the impression that recalling the definition of the dot-product will help you solve this problem. I strongly disagree. I was perfectly aware of the definition of the dot-product, but that did not help me in any way.In this case, I think it was reasonable to expect the students to see the dot, and recall the definition of the dot product. But if I had been in his shoes, I would have explained it, just in case.
You are given:
[tex]X=a_0 + \sigma \cdot a[/tex]
You are told that a0 is a number
You are told that a is a three-by-one vector
You are told that X is a 2x2 matrix
You are not told anything at all about the nature of σ.
The key to this problem is knowing the definition of σ. There's no way to do it if you don't know.