Draft:Singular matrix

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

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

Key properties and characteristics

• Determinant is zero: By definition an singular matrix have determinant of zero. Consequently, any co-factor expansion or determinant formula yields zero.
• Non-invertible: Since det(A)=0, the classic inverse does not exist in the case of singular matrix.
• Rank deficiency: Any structural reason that reduces the rank will cause singularity. For instance, if in a