The question from the image is:

"Let $A$ be a $5 \times 5$ real matrix whose sum of entries in each row is 2. Find the sum of entries of the matrix $A^3$."

Solution:

1. Let $A$ be a $5 \times 5$ matrix where the sum of the entries in each row is 2. Denote the row sum operator by $\mathbf{1}$, which is a column vector of all ones:

   1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}.$$

2. The row sums of $A$ can be expressed mathematically as: $A \cdot \mathbf{1} = 2 \cdot \mathbf{1}.$

3. Since this property propagates under matrix powers, for any power $n$, $A^n \cdot \mathbf{1} = 2^n \cdot \mathbf{1}.$

4. The sum of all entries of a matrix $A^n$ is given by the dot product: $\mathbf{1}^T \cdot (A^n \cdot \mathbf{1}).$

5. For $A^3$, substitute $A^3 \cdot \mathbf{1} = 2^3 \cdot \mathbf{1}$: $\mathbf{1}^T \cdot (A^3 \cdot \mathbf{1}) = \mathbf{1}^T \cdot (8 \cdot \mathbf{1}) = 8 \cdot \mathbf{1}^T \cdot \mathbf{1}.$

6. Since $\mathbf{1}^T \cdot \mathbf{1} = 5$ (there are 5 rows): $\text{Sum of entries of } A^3 = 8 \cdot 5 = 40.$

Final Answer:

The sum of the entries of $A^3$ is 40.

Would you like a further explanation, or do you have questions? Here are some related questions to explore:

1. What is the significance of row sums in determining the eigenvalues of a matrix?
2. Can this property generalize to non-square matrices?
3. How do we compute $A^n$ for a diagonalizable matrix?
4. What changes if the row sum is a value other than 2?
5. Can this technique apply to stochastic matrices or other specific matrix classes?

Tip: To simplify calculations with powers of matrices, eigenvalue-based approaches are often useful when the matrix is diagonalizable.