$ : A = \left[\begin{matrix}1 & 0 \\1 & 1 \\\end{matrix}\right]$
$ A^2 = \left[\begin{matrix}1 & 0 \\1 & 1 \\\end{matrix}\right]\left[\begin{matrix}1 & 0 \\1 & 1 \\\end{matrix}\right]= \left[\begin{matrix}1 & 0 \\2 & 1 \\\end{matrix}\right]$
$ A^3 = A^2 A = \left[\begin{matrix}1 & 0 \\2 & 1 \\\end{matrix}\right]\left[\begin{matrix}1 & 0 \\1 & 1 \\\end{matrix} \right]= \left[\begin{matrix}1 & 0 \\3& 1 \\\end{matrix}\right]$
ધારો કે $A^{n-1} = \left[\begin{matrix}1 & 0 \\n-1 & 1 \\\end{matrix}\right]$
$ A^{n-1} A = \left[\begin{matrix}1 & 0 \\n-1 & 1 \\\end{matrix}\right]\left[\begin{matrix}1 & 0 \\1 & 1 \\\end{matrix} \right]= \left[\begin{matrix}1 & 0 \\n & 1 \\\end{matrix}\right]$
હવે $ nA - (n-1) I = \left[\begin{matrix}n & 0 \\n & n \\\end{matrix}\right]\left[\begin{matrix}n-1 & 0 \\0 & n-1 \\\end {matrix}\right]= \left[\begin{matrix}1 & 0 \\n & 1 \\\end{matrix}\right] = A^n$
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