**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# Determinants and Row Echelon Form

Description

This lecture covers the concept of determinants and row echelon form, explaining how to calculate determinants using row operations and the significance of the row echelon form in determining the invertibility of a matrix. The lecture also explores the properties of determinants, such as their relationship with matrix operations and their geometric interpretation in terms of areas and volumes.

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

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

Related concepts (124)

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.

Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.

Related lectures (1,000)

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

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

Elementary Matrices and Inverses

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

Linear Applications: Vectorial Structure

Explores the vectorial structure of linear applications, including addition, scalar multiplication, and matrix representation.

Matrix Determinants: Properties and ApplicationsMATH-111(e): Linear Algebra

Covers the properties of matrix determinants and their geometric interpretation.

Characterization of Invertible Matrices

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