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If A and B are square matrices of same o...

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

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We have `AB=-BA`
`(A+B) (A^(2)-AB-B^(2))`
`=A^(3)-A^(2)B-AB^(2)+BA^(2)-BAB-B^(3)`
`=A^(3)-B^(3)-A^(2)B-AB^(2)-ABA+AB^(2)`
`( :' BA^(2)=BA A=-ABA and -BAB=AB B=AB^(2))`
`=A^(3)-B^(3)-A^(2)B+A^(2)B`
`=A^(3)-B^(3)`
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