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, ${\displaystyle det(A)=0}$. 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 ${\displaystyle rank(A)<n}$.

One of the basic condition of a singular matrix is that its determinant is equal to zero. If a matrix has determinant of zero, i.e. ${\displaystyle det(A)=0}$, then the columns ${\displaystyle C_{i}}$ are supposed to be linearly dependent. Determinant is an alternating multilinear form on columns, so any linear dependence among columns makes the determinant zero in magnitude. Hence ${\displaystyle det(A)=0}$.

For example:

if ${\displaystyle {\begin{bmatrix}1&2\\2&4\end{bmatrix}}}$

here ${\displaystyle C_{2}=2C_{1}}$, which implies that columns are linearly dependent.

An invertible matrix helps in the algorithm by providing an assumption that certain transformations, computations and systems can be reversed and solved uniquely, like ${\displaystyle Ax=B}$ to ${\displaystyle x=A^{-1}B}$. This helps solver to make sure if a solution is unique or not.

In Gaussian elimination, invertibility of the coefficient matrix ${\displaystyle A}$ ensures the algorithm produces a unique solution. For example, when matrix is invertible the pivots are non-zero, allowing one to row swap if necessary and solve the system, however in case of a singular matrix, some pivots can be zero which can not be fixed by mere row swaps. This imposes a problem where the elimination either breaks or gives an inconsistent result. One more problem a singular matrix produces when solving a Gaussian Elimination is that it can not solve the back substitution because to back substitute the diagonal entries of the ${\displaystyle A}$ matrix must be non-zero, i.e. ${\displaystyle det(A)\neq 0}$. However, in case of singular matrix the result is often infinitely many solutions.