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If A=[[0,1],[1,0]],B=[[0,-i],[i,0]] and ...

If `A=[[0,1],[1,0]],B=[[0,-i],[i,0]] and C=[[i,0],[0,-i]]`, show that `A^2=B^2=-C^2=I_2 and AB=-BA,AC=-CA and BC=-CB`.

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(i) if A=[{:(1,0),(0,1):}],B=[{:(0,1),(1,0):}]and C=[{:(1,0),(0,1):}], then show that A^(2)=B^(2)=C^(2)=I_(2). (ii) if A=[{:(1,0),(1,1):}],B=[{:(2,0),(1,1):}]and C=[{:(-1,2),(3,1):}], then show that A(B+C)=AB+AC. (iii) if A=[{:(1,-1),(-1,1):}]and B=[{:(1,1),(1,1):}], then show that AB is a zero matrix.