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Question
If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N
Solution
A and B are square matrices of the same order such that AB = BA.
To prove P(n) : AB" = B"A, `n in N`
For n = 1, we have:
Therefore, the result is true for n = 1.
Let the result be true for n = k.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have AB" = B"A, `n in N`
Now, we prove that (AB)" = A"B" for all n ∈ N
For n = 1, we have:
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have (AB)" = A"B", for all natural numbers.
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