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

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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)]

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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 An - 1

An - 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 = An – 1 × A