Let 'A' is `(4xx4)` matrix such that the sum of elements in each row is 1. Find out sum of the all the elements in `A^(10)`.

Answer

To solve the problem step-by-step, we can follow this approach:

### Step 1: Understand the Matrix Properties
Given that matrix \( A \) is a \( 4 \times 4 \) matrix where the sum of the elements in each row is 1, we can denote the rows of matrix \( A \) as:
- Row 1: \( A_{1,1}, A_{1,2}, A_{1,3}, A_{1,4} \)
- Row 2: \( A_{2,1}, A_{2,2}, A_{2,3}, A_{2,4} \)
- Row 3: \( A_{3,1}, A_{3,2}, A_{3,3}, A_{3,4} \)
- Row 4: \( A_{4,1}, A_{4,2}, A_{4,3}, A_{4,4} \)

The condition implies:
\[ A_{i,1} + A_{i,2} + A_{i,3} + A_{i,4} = 1 \quad \text{for } i = 1, 2, 3, 4 \]

### Step 2: Define a Vector
Define a vector \( M \) as:
\[ M = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} \]
This vector represents the sum of each row of \( A \).

### Step 3: Matrix Multiplication
When we multiply matrix \( A \) by vector \( M \):
\[ A \cdot M = \begin{pmatrix} A_{1,1} + A_{1,2} + A_{1,3} + A_{1,4} \\ A_{2,1} + A_{2,2} + A_{2,3} + A_{2,4} \\ A_{3,1} + A_{3,2} + A_{3,3} + A_{3,4} \\ A_{4,1} + A_{4,2} + A_{4,3} + A_{4,4} \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} = M \]

### Step 4: Inductive Argument
By induction, we can show that:
\[ A^n \cdot M = M \quad \text{for any natural number } n \]
This means that multiplying \( A \) by \( M \) repeatedly will always yield \( M \).

### Step 5: Calculate \( A^{10} \cdot M \)
Now, we need to find \( A^{10} \cdot M \):
\[ A^{10} \cdot M = M \]

### Step 6: Sum of Elements
To find the sum of all elements in \( A^{10} \), we can sum the elements of \( M \):
\[ \text{Sum of elements in } M = 1 + 1 + 1 + 1 = 4 \]

### Conclusion
Thus, the sum of all the elements in \( A^{10} \) is \( 4 \).

---