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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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`because A= B+CrArr A^(p+1) = (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)`
`= B^(p+1)+ ""^(p+1) C_(1) B^(p) C + 0 + 0 +... [because C^(2) = 0 rArr C^(2) = C^(3) =... = 0]`
`= B^(P) [B+(p+1) C]`
Hence, `A^(p+1) = B^(p)[B+(p+1)C]`
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