If A = [(1,1)(0,1)] , prove that A^n = [(1,n)(0,1)] for all n ∈ N.

Question

If A = \( \begin{bmatrix} 1 & 1 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix},\) prove that Aⁿ = \( \begin{bmatrix} 1 & n \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) for all n ∈ N.

A = [(1,1)(0,1)]

Aⁿ = [(1,n)(0,1)]

Answer 1

Given : A = \( \begin{bmatrix} 1&1 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix},\)

Matrix A is of order 2 x 2.

To prove : Aⁿ = \( \begin{bmatrix} 1&n \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\)

Proof :

A = \( \begin{bmatrix} 1&1 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\)

Let us assume that the result holds for A^{n - 1}

A^{n - 1} = \( \begin{bmatrix} 1& n-1 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}\)

We need to prove that the result holds for Aⁿ by mathematical induction .

Aⁿ = A^{n – 1} × A