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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

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