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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