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)`.

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

1.	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)`.
Answer» 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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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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