Singular matrix

A singular matrix is a square matrix whose determinant . Because the determinant is zero, the matrix cannot be inverted.

The following are all equivalent statements—i.e., they are true if and only if the others are true.

• The matrix is singular.
• There are either no nontrivial solutions or infinitely many solutions to .
• The inverse of does not exist.
• The matrix has linearly dependent rows or columns.
• The matrix has at least one row or column that can be expressed as a linear combination of the others.