Singular matrix

Common card · No. 334279

A singular matrix is a square matrix that is not invertible, unlike non-singular matrices which are invertible. Equivalently, an n-by-n matrix A is singular if and only if its determinant, 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 n-by-n matrix A is singular exactly when its columns (and rows) are linearly dependent, so that the linear map x↦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.

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