2: Matrices
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- 2.1: Matrix Operations
- You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications. In this section, we explore some matrix operations.
- 2.2: The Inverse of a Matrix
- This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row operations. It provides examples illustrating both the derivation of inverses and scenarios where matrices lack inverses.
- 2.3: Elementary Matrices
- This page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expresses a matrix as a product of elementary matrices. Properties of invertible matrices are discussed, including the conditions that an n×n matrix must meet to be invertible, emphasizing the significance of row operations.
- 2.4: LU Factorization
- An LU factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix L which has the main diagonal consisting entirely of ones, and an upper triangular matrix U in the indicated order.
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