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Question
Answer the following question:
If A = `[(3, -4),(1, -1)]`, prove that An = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N
Solution
Since the result to be proved for all n ∈ N, we will use the method of induction.
Let P(n) ≡ An = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`
If n = 1, then A = `[(3, -4),(1, -1)]`
which is given
∴ P(1) is true.
Assume that P(n) is true for n = k
i.e., Ak = `[(1 + 2"k", -4"k"),("k", 1 - 2"k")]` ...(1)
To prove that P(n) is true for n = k + 1
i.e., to prove that,
Ak+1 = `[(1 + 2("k" + 1), -4("k" + 1)),("k" + 1, 1-2("k" + 1))]`
= `[(2"k" + 3, -4"k" - 4),("k" + 1, -2"k" - 1)]`
L.H.S. = Ak+1 = Ak·A
= `[(1 + 2"k", -4"k"),("k", 1 - 2"k")] [(3, -4),(1, -1)]` ...[By (1)]
= `[((1 + 2"k")3 + (-4"k")(1), (1 + 2"k")(-4)+(-4"k")(-1)),(3"k" + (1 - 2"k")(1), "k"(-4) + (1 - 2"k")(-1))]`
= `[(2"k" + 3, -4"k" - 4),("k" + 1, -2"k" - 1)]`
= R.H.S.
∴ if P(n) is true for n = k, then it is also true for n = k + 1. Hence, by the method of induction P(n) is true for all n ∈ N.
i.e., An = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N.
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