--- $\textbf{Exercise 13 Commentary:}$ This exercise provides an explicit formula for the $(j, k)$-entry of the $n$-th power $A^n$ of an $n \times n$ matrix $A$, in terms of the entries of $A$. Specifically, it shows that $(A^n)_{j,k} = \sum_{i_1, i_2, \ldots, i_n = 1}^n A_{j, i_1}A_{i_1, i_2} \cdots A_{i_{n-1}, i_n}A_{i_n, k}$. The formula is derived by considering the definition of matrix multiplication and expanding the entries of the product $A^n = A \cdots A$ using the entries of $A$. It highlights how the $(j, k)$-entry of $A^n$ depends on all possible length-$n$ products of entries from $A$, tracing paths from row $j$ to column $k$ through intermediate rows and columns. $\textbf{Exercise 13 Examples:}$ 1) Let $A = \begin{pmatrix}1&2\\3&4\end{pmatrix}$. Then $A^2 = \begin{pmatrix}7&10\\15&22\end{pmatrix}$, and the formula gives $(A^2)_{1,1} = 1 \cdot 1 + 2 \cdot 3 = 7$, $(A^2)_{1,2} = 1 \cdot 2 + 2 \cdot 4 = 10$, $(A^2)_{2,1} = 3 \cdot 1 + 4 \cdot 3 = 15$, $(A^2)_{2,2} = 3 \cdot 2 + 4 \cdot 4 = 22$. 2) Let $A = \begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}$. Then $A^3 = \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} = I_3$, and the formula correctly gives $(A^3)_{j,k} = 1$ if $j = k$, and 0 otherwise. 3) Let $A$ be the $4 \times 4$ matrix with $1s on the main diagonal and $2s everywhere else. Then $(A^2)_{j,k} = 1 \cdot 2 \cdot 1 + 2 \cdot 2 \cdot 2 + 2 \cdot 2 \cdot 2 + 2 \cdot 2 \cdot 1 = 18$ for all $j \neq k$, and $(A^2)_{j,j} = 1 \cdot 1 \cdot 1 + 2 \cdot 2 \cdot 2 + 2 \cdot 2 \cdot 2 + 2 \cdot 2 \cdot 1 = 22$. $\textbf{Implications and Applications:}$ This formula provides an explicit way to compute higher powers of a matrix directly from its entries, without needing to perform successive matrix multiplications. It is useful for studying the long-term behavior of discrete dynamical systems governed by matrix powers, such as Markov chains. The formula also connects to the theory of matrix functions, since matrix powers are special cases of more general matrix functions like the matrix exponential. From a purely algebraic perspective, it sheds light on the multiplicative structure of matrices and their representations of certain algebraic structures like groups and algebras. --- $\textbf{Exercise 13 Implications and Applications (continued):}$ The explicit formula for the entries of matrix powers provided in this exercise has important implications in the study of discrete dynamical systems, Markov chains, and the analysis of matrix functions. In the context of discrete dynamical systems, the evolution of the system over time is often described by a sequence of matrices, where each matrix represents the system's behavior during a particular time step. The matrix powers arise naturally in this setting, as they describe the system's behavior over multiple time steps. The formula presented in this exercise allows for efficient computation of these matrix powers, which is crucial for analyzing the long-term behavior and stability properties of the dynamical system. Markov chains are widely used in various fields, including probability theory, statistical mechanics, and machine learning. In Markov chain analysis, the transition matrix governs the evolution of the system from one state to another. The matrix powers of the transition matrix provide valuable information about the long-term behavior of the Markov chain, such as the existence and properties of stationary distributions. The formula in this exercise enables efficient computation of these matrix powers, facilitating the analysis of complex Markov chain models. In the study of matrix functions, matrix powers are special cases of more general matrix functions, such as the matrix exponential and the matrix logarithm. These matrix functions play important roles in various applications, including the analysis of continuous dynamical systems, numerical integration of differential equations, and the study of matrix groups and Lie algebras. The explicit formula for matrix powers can serve as a building block for developing efficient algorithms and approximation techniques for computing more general matrix functions. Furthermore, the formula has applications in the study of matrix representations of algebraic structures, such as groups and algebras. In these contexts, the matrix powers encode important information about the structure and properties of the underlying algebraic object, and the ability to efficiently compute them can provide insights into the behavior and characteristics of these structures. Overall, the explicit formula for the entries of matrix powers is a powerful tool that has applications in a wide range of fields, including dynamical systems, probability theory, numerical analysis, and abstract algebra, enabling efficient computations and facilitating the analysis of complex systems and algebraic structures.