If `B ,\ C` are `n` rowed square matrices and if `A=B+C` , `B C=C B` , `C^2=O` , then show that for every `n in N` , `A^(n+1)=B^n(B+(n+1)C)` .

Explanation

Given that,

`A= B+C`

`implies A^2=(B+C)(B+C) `

`=B(B+C)+C(B+C) `

`=B^{2}+B C+C B+C^{2} `

`=B^{2}+2 B C ` because`[ B C=CB text { and } C^{2}=O] `

`=B(B+2 C) `

It is true for `n=1`.

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