If $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$, prove that $A^{n}=\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]$ for all positive integers $n$.
We shall prove the result by the principle of mathematical induction on n.
Step 1: If n = 1, by definition of integral powers of matrix, we have
$A^{1}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]=A$
So, the result is true for $n=1$.
Step 2: Let the result be true for $n=m$. Then,
$A^{m}=\left[\begin{array}{cc}1 & m \\ 0 & 1\end{array}\right]$ ...(1)
Now, we shall show that the result is true for $n=m+1$.
Here,
$A^{m+1}=\left[\begin{array}{cc}1 & m+1 \\ 0 & 1\end{array}\right]$
By definition of integral power of matrix, we have
$A^{m+1}=A^{m} A$
$=\left[\begin{array}{cc}1 & m \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ $[$ From eq. $(1)]$
$=\left[\begin{array}{cc}1+0 & 1+m \\ 0+0 & 0+1\end{array}\right]$
$=\left[\begin{array}{cc}1 & 1+m \\ 0 & 1\end{array}\right]$
This shows that when the result is true for n = m, it is also true for n = m + 1.
Hence, by the principle of mathematical induction, the result is valid for any positive integer n.