The determinant of a matrix is a scalar value that provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation it represents. A zero determinant indicates that the matrix is singular, meaning it does not have an inverse and its rows or columns are linearly dependent.

Determinant Of A Matrix

determinant of a matrix

scalar value

important information

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.

Invertible Matrix

volume scaling factor

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication, mapping lines to lines or points through the origin. These transformations can be represented by matrices, making them fundamental in solving systems of linear equations and understanding geometric transformations in higher dimensions.

linear transformation

zero determinant

A singular matrix is a square matrix that does not have an inverse, characterized by a determinant of zero. This property implies that its rows or columns are linearly dependent, meaning they do not span the entire vector space.

Singular Matrix

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.

Relevant Degrees

Your Lessons