QM - Spin operator conjugate question

[QMQuestions2]

Homework Statement

Okay so I've got a question I really need answered first up! If I have a 2x1 matrix for Psi, is Psi* a 1x2 matrix with all the 'i's turned to '-i's?

Now onto the actual question - http://imgur.com/3ucb4" - part b only

Homework Equations

http://imgur.com/bcEm3"

(Sorry to URL everything)

The Attempt at a Solution

http://imgur.com/FW0dP"

What did I do wrong?

 

[Unto]

Simply have the 2 x 2 matrix of Sx operate on the state v, then multiply your new 2 x 2 matrix with the conjugate of v.

And yes the conjugate of v is a transpose matrix with the i's all different.

 

[QMQuestions2]

Isn't that what I did? v is a 2x1 matrix. When I multiply Sx with v I'm left with a 2x1 matrix, not a 2x2.

Sx times v = v

v* times v = 2

Integral = infinity