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If A and B are square matrices such that...

If A and B are square matrices such that `AB = BA` then prove that `A^(3)-B^(3)=(A-B) (A^(2)+AB+B^(2))`.

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`(A-B)(A^(2)+AB+B^(2))`
`=A^(3)+A^(2)B+AB^(2)-BA^(2)-BAB-B^(3)`
Now `A^(2)B=A(AB)=A(BA)=(AB)A=BA A=BA^(2)`
And `AB^(2)=(AB)B=(BA)B`
`:. (A-B) (A^(2)+AB+B^(2))=A^(3)-B^(3)`
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