What does it indicate when the determinant of a matrix is zero?

When the determinant of a matrix is zero, it signifies that the matrix is singular and does not have an inverse. This concept is crucial in linear algebra. A matrix is said to be singular when it does not have full rank, meaning that the columns (or rows) of the matrix are linearly dependent. This dependency implies that at least one column can be expressed as a linear combination of others, leading to the loss of unique solutions for the associated linear system.

In practical terms, if a matrix has a zero determinant, any attempt to find its inverse will fail since the inverse only exists for non-singular (or invertible) matrices. This condition is often tied to geometric interpretations as well; for example, in a two-dimensional space, a matrix representing a transformation with a zero determinant transforms space in such a way that it collapses volume into a lower dimension (e.g., compressing a 2D area into a line).

While the other options pivot around different properties of matrices, they do not directly pertain to the implication of a zero determinant. An inverse only exists when the determinant is non-zero, symmetry doesn't imply anything about the determinant being zero, and complex eigenvalues are unrelated to this specific determinant condition within the context discussed.