A is n by n.
| Nonsingular | Singular |
|---|---|
| A is invertible. | A is not invertible. |
| The columns are independent. | The columns are dependent. |
| The rows are independent. | The rows are dependent. |
| The determinant is not zero. | The determinant is zero. |
| Ax = 0 has one solution x = 0. | Ax = 0 has infinitely many solutions. |
| Ax = b has one solution x = A⁻¹b. | Ax = b has no solution or infinitely many. |
| A has n (nonzero) pivots. | A has r < n pivots. |
| A has full rank r = n. | A has rank r < n. |
| The reduced row echelon form is R = I. | R has at least one zero row. |
| The column space is all of Rⁿ. | The column space has dimension r < n. |
| The row space is all of Rⁿ. | The row space has dimension r < n. |
| All eigenvalues are nonzero. | Zero is an eigenvalue of A. |
| AᵀA is symmetric positive definite. | AᵀA is only semidefinite. |
| A has n (positive) singular values. | A has r < n singular values. |
Each line of the singular column can be made quantitative using row.