Define a determinant in the context of matrices.

A determinant is a scalar value that is particularly significant in linear algebra and is used to characterize certain properties of a matrix. It serves several key functions, one of which is to determine whether a matrix is invertible. If the determinant of a square matrix is zero, this indicates that the matrix does not have an inverse and is singular; conversely, a non-zero determinant indicates that the matrix is invertible.

Beyond invertibility, the determinant provides insights into the geometric interpretation of transformations represented by the matrix, such as scaling and rotation. For instance, in two dimensions, the absolute value of the determinant represents the area scaling factor, and in three dimensions, it represents the volume scaling factor.

While other choices mention aspects related to the determinant, they do not encapsulate its comprehensive definition as a measure of properties like invertibility, which is fundamental in both theoretical aspects of linear algebra and practical applications in solving systems of equations and transformations.