The matrix inverse is a fundamental concept in linear algebra, representing a matrix that, when multiplied by the original matrix, yields the identity matrix. Not all matrices have inverses, and a matrix must be square and have a non-zero determinant to be invertible.

Matrix Inverse

matrix inverse

fundamental concept

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.

linear algebra

original matrix

An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere, serving as the multiplicative identity in matrix algebra. This means that when any matrix is multiplied by an identity matrix of compatible dimensions, the original matrix is unchanged, analogous to multiplying a number by one in arithmetic.

identity matrix

A square matrix is a matrix with the same number of rows and columns, often used to represent linear transformations in vector spaces. Its properties, such as determinant, eigenvalues, and trace, are foundational in linear algebra and have significant applications in various scientific fields.

Square Matrix

non-zero determinant

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. The existence of an inverse is equivalent to the matrix having a non-zero determinant and full rank, ensuring that the linear transformation it represents is bijective.

Relevant Degrees

Your Lessons