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If B ,C are square matrices of order na ...

If `B ,C` are square matrices of order `na n difA=B+C ,B C=C B ,C^2=O ,` then without using mathematical induction, show that for any positive integer `p ,A^(p-1)=B^p[B+(p+1)C]` .

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`A^(p+1)=(B+C)^(p+1)`
Since `BC=CB`, we have
`(B+C)^(p+1)`
`=^(p+1)C_(0) B^(p+1)+^(p+1)C_(1)B^(p)C+^(p+1) C_(2)B^(p-1) C^(2)+...+^(p+1)C_(p+1)C^(p+1)`
`=^(p+1)C_(0)B^(p+1)+^(p+1)C_(1)B^(p) C+O+O+...+O" "( :' C^(2)=O implies C^(3)=C^(4)=... =O)`
`=B^(p+1)+(p+1) B^(p)C`
`=B^(p)[B+(p+1)C]`
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