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Question
If A is a square matrix, using mathematical induction prove that (AT)n = (An)T for all n ∈ ℕ.
Solution
Let the given statement P(n), be given as
P(n): (AT)n = (An)T for all n ∈ ℕ.
We observe that
P(1): (AT)1 = AT = (A1)T
Thus, P(n) is true for n = 1.
Assume that P(n) is true for n = k ∈ ℕ.
i.e., P(k): (AT)k = (Ak)T
To prove that P(k + 1) is true, we have
(AT)k + 1 = (AT)k.(AT)1
= (Ak)T.(A1)T
= (Ak + 1)T
Thus, P(k + 1) is true, whenever P(k) is true.
Hence, by the Principle of mathematical induction, P(n) is true for all n ∈ ℕ.
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