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If det, (A-B) ne 0, A^(4)=B^(4), C^(3) A...

If det, `(A-B) ne 0, A^(4)=B^(4), C^(3) A=C^(3)B` and `B^(3)A=A^(3)B`, then find the value of det. `(A^(3)+B^(3)+C^(3))`.

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`(A^(3)+B^(3)+C^(3)) (A-B)`
`=A^(4)-A^(3)B+B^(3)A-B^(4)+C^(3)A-C^(3) B=O`
`:.` det. `((A^(2)+B^(3)+C^(3))(A-B))=0`
`implies` det. `(A^(3)+B^(3)+C^(3))=0`, as det. `(A-B) ne 0`
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