if B, C ARE n ROWED SQUARE MATRICES AND IF A=B+C , BC=CB ,C2=0 ,THEN SHOW THAT FOR EVERY n is the element ofN, A n+1 =Bn(B+(n+1)C).

Let P(n) be the statement given by

For n = 1, we have


Hence, the statement is true for n = 1.

Assume that the statement is true for n = k.

We shall now show that P(k + 1) is true. That is we need to show that

Now,

So, the statement is true for n = k + 1.

Thus, P(k) is true ÞP(k + 1) is true.

Hence, by the principle of mathematical induction P (n) is true for all n ÎN.

 

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