If A=[{:(1,2),(-2,1):}],B=[{:(2,3),(3,-4...

If `A=[{:(1,2),(-2,1):}],B=[{:(2,3),(3,-4):}]` and `C=[{:(1,0),(-1,0):}]`, verfity (i) A(B+C)=AB+AC.

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We have, `A=[{:(1,2),(-2,1):}]B=[{:(2,3),(3,-4):}]` and `C=[{:(1,0),(-1,0):}]`
(i) `(AB)=[{:(1,2),(-2,1):}][{:(2,3),(3,-4):}]=[{:(2+6,3-8),(-4+3,-6-4):}]=[{:(8,-5) ,(-1-,-10):}]`
and `(AB)C=[{:(8,-5),(-1,-10),(-1 ,- 10):}][{:(1,0),(-1,0):}]`
`=[{:( 8+5,0),(-1,+10,0):}]=[{:(13,0),(9,0):}]`
Again, `(BC)=[{:(2,3),(3,-4):}][ {:(-1,0),(7,0):}]`
and `A(BC)=[{:(1,2),(-2,1):}][{:(-1,0),(7,0):}]`
`=[{:(-1+14,0),(+2+7,0):}]=[{:(13,0),(9,0):}]`
`therefore (AB)C=A(BC)`
(ii) `(B+C)=[{:(2,3),(3,-4):}]+[{:(1,0),(-1,0):}]=[{:(3,3),(2,-4):}]`
and `AB=[{:( 1,2),(-2,1):}][{:(3,3),(2,-4):}]`
`=[{:(3+4,3-8),(-6+2,-6-4):}][{:(2,3),(3,-4):}]`
`=[{:(2+6 ,3-8),(-4+3,-6-4):}]=[{:(8,-6),(-1,-10):}] `
and `AC=[{:( 1,2),(-2,1):}][{:(1, 0),(-1,0):}]=[{:(1-2,0),(-2-1,0):}]=[{:(-1,0),(-3,0):}]`
`therefore AB+AC=[{:( 8,-5),(- 1,-10):}]+[{:(-1,0),(-3,0):}]`
`rArr AB+AC=[{:(7,-5),(-4,-10):}]`
From Eqs. iii and iv
`A(B+C)=AB+AC`

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