**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Matrix Operations: Determinants and Inverses

Description

This lecture covers the properties of matrices, focusing on determinants and inverses. The instructor explains how to calculate determinants, the properties of the inverse of a matrix, and demonstrates the application of these concepts through various examples.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

Instructor

Related concepts (23)

MATH-111(pi): Linear algebra (flipped classroom)

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. Cette classe pilote est donné sous forme inversée.

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form.

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or .

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries above the main diagonal are zero. Similarly, a square matrix is called if all the entries below the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.

In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If is an matrix, where is the entry in the -th row and -th column of , the formula is where is the sign function of permutations in the permutation group , which returns and for even and odd permutations, respectively. Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes which may be more familiar to physicists.

Related lectures (32)

Matrix Operations: Inverse and Reduction to Echelon FormMATH-111(e): Linear Algebra

Covers matrix operations and reduction to echelon form with practical examples.

Characteristic Polynomials and Similar MatricesMATH-111(e): Linear Algebra

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.

Matrix Powers and Inverse: Examples and ApplicationsMATH-111(c): Linear Algebra

Covers examples of symmetric and anti-symmetric matrices, matrix powers, and the concept of matrix inverse.

Characterization of Invertible Matrices

Explores the properties of invertible matrices, including unique solutions and linear independence.

Elementary Matrices and Inverses

Covers the concept of elementary matrices and their role in obtaining the inverse of a matrix.