If `A=[{:(,1,a),(,0,1):}]` then find `lim_(n-oo) (1)/(n)A^(n)`

Answer

To solve the problem, we need to find the limit:

\[ \lim_{n \to \infty} \frac{1}{n} A^n \]

where \( A = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \).

### Step 1: Calculate \( A^2 \)

First, we calculate \( A^2 \):

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \]

Using matrix multiplication:

\[ A^2 = \begin{pmatrix} 1 \cdot 1 + a \cdot 0 & 1 \cdot a + a \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot a + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 2a \\ 0 & 1 \end{pmatrix} \]

### Step 2: Calculate \( A^3 \)

Next, we calculate \( A^3 \):

\[ A^3 = A^2 \cdot A = \begin{pmatrix} 1 & 2a \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \]

Using matrix multiplication:

\[ A^3 = \begin{pmatrix} 1 \cdot 1 + 2a \cdot 0 & 1 \cdot a + 2a \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot a + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 3a \\ 0 & 1 \end{pmatrix} \]

### Step 3: Generalize \( A^n \)

From the calculations of \( A^2 \) and \( A^3 \), we can see a pattern emerging. We can generalize this to:

\[ A^n = \begin{pmatrix} 1 & na \\ 0 & 1 \end{pmatrix} \]

### Step 4: Substitute \( A^n \) into the limit expression

Now, substituting \( A^n \) into the limit expression:

\[ \frac{1}{n} A^n = \frac{1}{n} \begin{pmatrix} 1 & na \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{n} & a \\ 0 & \frac{1}{n} \end{pmatrix} \]

### Step 5: Take the limit as \( n \to \infty \)

Now, we take the limit as \( n \to \infty \):

\[ \lim_{n \to \infty} \begin{pmatrix} \frac{1}{n} & a \\ 0 & \frac{1}{n} \end{pmatrix} = \begin{pmatrix} 0 & a \\ 0 & 0 \end{pmatrix} \]

### Final Result

Thus, the final result is:

\[ \lim_{n \to \infty} \frac{1}{n} A^n = \begin{pmatrix} 0 & a \\ 0 & 0 \end{pmatrix} \]