What property does the determinant of a matrix describe?

The chosen answer clearly states that the determinant describes both the scaling factor of a transformation and matrix invertibility, which are key characteristics associated with determinants in linear algebra.

The determinant can be interpreted as a scaling factor that describes how the transformation represented by the matrix affects area (in two dimensions), volume (in three dimensions), or more generally, n-dimensional content in higher dimensions. Specifically, if a matrix has a determinant of 2, it indicates that the transformation doubles the size of areas or volumes. Conversely, a determinant of 0 suggests that the transformation collapses the space into a lower dimension, essentially indicating that the linear transformation is not invertible.

Furthermore, the determinant is directly related to a matrix's invertibility: a matrix is invertible if and only if its determinant is non-zero. When the determinant is zero, it indicates that the matrix does not have a full rank, leading to a lack of unique solutions for the associated linear transformation or equation.

Therefore, this property of the determinant encapsulates both the nature of the transformation it describes and the conditions under which the matrix can be inverted, making the selection accurate in the context of what the determinant signifies in linear algebra.