Answer :

Given A is a square matrix.

We need to prove that (AT)n = (An)T.

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

Step 1: When n = 1, we have (AT)1 = AT

(AT)1 = (A1)T

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.

(AT)k = (Ak)T

To prove the given equation using mathematical induction, we have to show that (AT)k+1 = (Ak+1)T.

We know (AT)k+1 = (AT)k × AT.

(AT)k+1 = (Ak)T × AT

We have (AB)T = BTAT.

(AT)k+1 = (AAk)T

(AT)k+1 = (A1+k)T

(AT)k+1 = (Ak+1)T

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, (AT)n = (An)T for all n ϵ N.

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