Which statement correctly identifies a singular matrix?

Which statement correctly identifies a singular matrix?

• A. It has an inverse
• B. Its determinant is nonzero
• C. It does not have a full rank
• D. It is always symmetric

Answer: (C)

A matrix is singular when it cannot be inverted, which happens exactly when its rows or columns are linearly dependent. That means its rank is less than its size, so it does not have full rank. Therefore the statement that correctly identifies a singular matrix is the one saying it does not have a full rank. If the determinant were nonzero, the matrix would be invertible, so it would not be singular. Symmetry has nothing to do with singularity, since a matrix can be singular without being symmetric (and can be symmetric and non-singular as well). A quick example is a 2×2 matrix whose second column is a multiple of the first; its determinant is zero and its rank is 1, so it is singular.

A matrix is singular when it cannot be inverted, which happens exactly when its rows or columns are linearly dependent. That means its rank is less than its size, so it does not have full rank. Therefore the statement that correctly identifies a singular matrix is the one saying it does not have a full rank.

If the determinant were nonzero, the matrix would be invertible, so it would not be singular. Symmetry has nothing to do with singularity, since a matrix can be singular without being symmetric (and can be symmetric and non-singular as well). A quick example is a 2×2 matrix whose second column is a multiple of the first; its determinant is zero and its rank is 1, so it is singular.