Q. 624.7( 3 Votes )

# If A = diag

Answer :

Given .

We need to prove that .

We will prove this result using the principle of mathematical induction.

Step 1: When n = 1, we have   Hence, the equation is true for n = 1.

Step 2: Let us assume the equation true for some n = k, where k is a positive integer. To prove the given equation using mathematical induction, we have to show that .

We know Ak+1 = Ak × A.   Ak+1 = diag(ak+1 bk+1 ck+1)

Hence, the equation is true for n = k + 1 under the assumption that it is true for n = k.

Therefore, by the principle of mathematical induction, the equation is true for all positive integer values of n.

Thus, An = diag(an bn cn) for all positive integers n.

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