Prove that spin in arbitary direction is hbar/2

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The discussion centers on proving that spin in an arbitrary direction is hbar/2. Participants suggest focusing on the eigenvalues of the spin matrix S_{z'} rather than S_{z'}^2. There is a clarification that the inquiry is about the spin component in a specific direction, not the total spin angular momentum S^2. Misunderstandings regarding the relationship between these eigenvalues are noted. The conversation emphasizes the importance of correctly identifying the relevant spin matrices for the proof.
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the question is in the jpeg file.
 

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How come you don't just solve the eigenvalues for the matrix that you have denoted by S_{z'}? We are not interested in the eigenvalues of S_{z'}^2, but just on the eigenvalues of the spin matrix you have constructed in the fourth line of your attachment. I think you have got a bit messed up with the eigenvalues related to the total spin angular momentum S^2=S_x^2+S_y^2+S_z^2 and the component of spin angular momentum related to an arbitrary direction.

I hope this made any sense to you, it's my first post!
 
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