If A=pmatrix1&01&1pmatrix and I=pmatrix1&00&1pmatrix, then which one of the following hold…

Question

If $A=\begin{pmatrix}1&0\1&1\end{pmatrix}$ and $I=\begin{pmatrix}1&0\0&1\end{pmatrix}$, then which one of the following holds for all $n\ge1$, by the principle of mathematical induction:

Accepted Answer

By induction, $A^n=\begin{pmatrix}1&0\n&1\end{pmatrix}$.

$nA-(n-1)I=\begin{pmatrix}n&0\n&n\end{pmatrix}-\begin{pmatrix}n-1&0\0&n-1\end{pmatrix}=\begin{pmatrix}1&0\n&1\end{pmatrix}=A^n$.

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