If A=[{:( 0,-x),(x,0):}].B=[{:(0,1),(1,0...

If `A=[{:( 0,-x),(x,0):}].B=[{:(0,1),(1,0):}]` and `x^(2)=-1` , then show that `(A+B)^(2)=A^(2)+B^(2)`.

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We have `A=[{:(0,-x),(x,0):}] B=[{:(0,1),(1,0):}]` and `x^(2)=-1`
`therefore (A+B)=[{:(0,-2+1),( x+1,0):}]`
and `(A+B)^(2)[{:(0,-x+1),(x+1,0):}][{:(0,-x+1),(x+ 1,0):}]`
`=[{:(1-x^(2),0),( 0,1-x^(2)):}] `
Also, `A^(2)=A.A=[ {:(0,-x),(x,0):}][{:(0,-x),(x,0):}]=[{:(-x^(2),0),(0,-x^(2)):}]`
and `B^(2)=B.B=[{:( 0,1) ,(1,0):}][{:(0,1),(1,0):}]=[{:(1,0),(0,1):}]`
Now, `A^(2)+B^(2)=[{:(-x^(2)+1,0),(0,-x^(2)+1):}]=[{:(1-x^(2),0),(0,1-x^(2)):}]`
`=(A+B)^(2)`

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