Let `A` be a `(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, we need to find the sum of all elements in the matrix \( A^{10} \), given that the sum of the elements in each row of the matrix \( A \) is 1.

### Step-by-Step Solution:

1. **Define the Matrix \( A \)**:
Let \( A \) be a \( 4 \times 4 \) matrix such that the sum of the elements in each row is 1.

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

3. **Calculate \( AM \)**:
When we multiply the matrix \( A \) by the vector \( M \), we get:
\[ AM = \begin{pmatrix} \text{sum of row 1} \\ \text{sum of row 2} \\ \text{sum of row 3} \\ \text{sum of row 4} \end{pmatrix} \]
Since the sum of the elements in each row of \( A \) is 1, we have:
\[ AM = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} = M \]

4. **Calculate \( A^2M \)**:
Now, consider \( A^2M \):
\[ A^2M = A(AM) = A(M) = M \]
This shows that \( A^2M = M \).

5. **Generalize for \( A^nM \)**:
By induction, we can show that for any natural number \( n \):
\[ A^nM = M \]
Therefore, specifically for \( n = 10 \):
\[ A^{10}M = M \]

6. **Sum of Elements in \( A^{10} \)**:
The sum of all elements in \( A^{10} \) can be found by summing the elements of \( M \):
\[ \text{Sum of elements in } A^{10} = 1 + 1 + 1 + 1 = 4 \]

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