Determinants are mathematical expressions used in linear algebra to provide important information about a square matrix. They can indicate whether a matrix is invertible, meaning it has an inverse, and help in solving systems of linear equations. The value of a determinant can be calculated using various methods, including expansion by minors or row reduction.
In geometry, determinants can also represent the area or volume of shapes defined by vectors. For example, the determinant of a 2x2 matrix can give the area of a parallelogram formed by two vectors, while a 3x3 matrix can represent the volume of a parallelepiped.