Singular matrix

A singular matrix is a square matrix that is not invertible, unlike non-singular matrix which is invertible. Equivalently, an ${\displaystyle n}$-by-${\displaystyle n}$ matrix ${\displaystyle A}$ is singular if and only if determinant, ${\displaystyle det(A)=0}$.^[1] In classical linear algebra, a matrix is called non-singular (or invertible) when it has an inverse; by definition, a matrix that fails this criterion is singular. In more algebraic terms, an ${\displaystyle n}$-by-${\displaystyle n}$ matrix A is singular exactly when its columns (and rows) are linearly dependent, so that the linear map ${\displaystyle x\rightarrow Ax}$ is not one-to-one.

In this case the kernel (null space) of A is non-trivial (has dimension ≥1), and the homogeneous system ${\displaystyle Ax=0}$ admits non-zero solutions. These characterizations follow from standard rank-nullity and invertibility theorems: for a square matrix A, ${\displaystyle det(A)\neq 0}$ if and only if ${\displaystyle rank(A)=n}$, and ${\displaystyle det(A)=0}$ if and only if ${\displaystyle rank(A)<n}$.