This paper, authored by Lenka Čížková and Pavel Čížek, discusses numerical linear algebra, focusing on matrix decompositions and their applications in solving systems of linear equations and matrix inversions. The authors introduce several key matrix decompositions, including Cholesky, LU, QR, and singular value decomposition (SVD), and explain how these decompositions facilitate numerically stable algorithms. They also cover direct methods for solving linear systems, such as Gauss-Jordan elimination and iterative refinement, and iterative methods, which construct a series of approximations that converge to the solution. The paper emphasizes the importance of these techniques in computational statistics and provides an overview of their implementation and numerical stability. Additionally, it addresses the challenges of solving large, sparse systems and the use of matrix decompositions to transform nearly singular problems into nonsingular ones. The authors conclude by highlighting the relationship between solving linear systems and computing matrix inverses, and they discuss the advantages and disadvantages of different methods in practical applications.This paper, authored by Lenka Čížková and Pavel Čížek, discusses numerical linear algebra, focusing on matrix decompositions and their applications in solving systems of linear equations and matrix inversions. The authors introduce several key matrix decompositions, including Cholesky, LU, QR, and singular value decomposition (SVD), and explain how these decompositions facilitate numerically stable algorithms. They also cover direct methods for solving linear systems, such as Gauss-Jordan elimination and iterative refinement, and iterative methods, which construct a series of approximations that converge to the solution. The paper emphasizes the importance of these techniques in computational statistics and provides an overview of their implementation and numerical stability. Additionally, it addresses the challenges of solving large, sparse systems and the use of matrix decompositions to transform nearly singular problems into nonsingular ones. The authors conclude by highlighting the relationship between solving linear systems and computing matrix inverses, and they discuss the advantages and disadvantages of different methods in practical applications.