What does it mean if a matrix's determinant equals zero?

When a matrix's determinant is equal to zero, it signifies that the matrix is non-invertible, also referred to as singular. This means that there is no unique way to reverse the transformation represented by the matrix; hence, it cannot be inverted. In practical terms, this implies that the linear system associated with the matrix does not have a unique solution.

This concept ties into the geometric interpretation of determinants as well: a determinant of zero indicates that the matrix either compresses space or maps vectors into a lower-dimension space, meaning that it collapses the volume (or area in two dimensions) spanned by its rows or columns. As a result, the rows or columns of the matrix are linearly dependent, failing to span the entire space.

As for the other options, they do not accurately describe the implications of a zero determinant. A unique matrix is related to invertibility, but a determinant of zero indicates the opposite. Infinite solutions pertain to the structure of solutions in a corresponding linear system but do not define the matrix itself. Lastly, orthogonality pertains to the relationship between the rows or columns being perpendicular to each other, which does not directly relate to the determinant being zero. Thus, the assertion that the matrix is