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

Answer 1

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., A^k = `[(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,

A^k+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. = A^k+1 = A^k·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.