An $n\times n$ <square matrix> $A$ over a <field>, with $n\geq1$, is singular when it has no <matrix inverse>. Geometrically, its <linear map> loses at least one input direction: some nonzero <vector> $v$ satisfies $Av=0$, so $A(x+v)=Ax$. Over the <real numbers>, the following family makes that loss visible:
$$
A(t)=\begin{pmatrix}1&0\\0&t\end{pmatrix},\qquad A(t)(x,y)=(x,ty).
$$

\Image[/singularity-nullity-collapse.png]
{title=Compression by $A(t)=\operatorname{diag}(1,t)$ becomes collapse only at $t=0$}
{description=The <square> $[-1,1]^2$ retains two-dimensional extent for $t=1$ and $t=0.15$, but maps to a segment at $t=0$. The orange points come from three inputs with the same first coordinate and different second coordinates. They coincide only at $t=0$. This is exact illustrative geometry, not measured data.}

For this <diagonal matrix>, $\det A(t)=t$. Whenever $t\ne0$, the input can be recovered as $(x,y)=(b_1,b_2/t)$ and the <kernel> is $\{0\}$. At $t=0$, the <kernel> becomes $\operatorname{span}\{(0,1)\}$: the entire second coordinate is forgotten. Consequently $A(0)x=(b_1,b_2)$ has no solution if $b_2\ne0$, and has the solution <straight line> $x=(b_1,s)$, $s\in\mathbb R$, if $b_2=0$. A <singular matrix> therefore need not make a particular <linear system> inconsistent; it prevents unique recovery when that <linear system> is consistent.

For an $n\times n$ <square matrix>, all these tests describe the same failure:
$$
\boxed{
A\text{ singular}
\iff\det A=0
\iff\operatorname{rank}A<n
\iff\ker A\ne\{0\}
\iff\operatorname{im}A\ne\mathbb F^n.
}
$$
Here $\mathbb F$ is the <field> of <scalars>. Equivalently, the columns are <linearly dependent>, the <linear map> is neither <injective> nor <surjective>, and some outputs cannot be reached. See the <invertible matrix> equivalences and https://www.ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/1963da71c4d96e5d14e7939780f79bcc_MIT18_06SCF11_Ses1.3sum.pdf[Strang's MIT 18.06SC lecture summary, Multiplication and inverse matrices, p. 2]. Over an infinite <field>, every consistent <linear system> for a <singular matrix> has infinitely many solutions; over a finite <field> with $q$ elements and <nullity> $d$, it has exactly $q^d$ solutions. This follows by adding each <kernel> <vector> to one solution.

The square hypothesis is essential. For an $m\times n$ <matrix>, <injectivity> means <matrix rank> $n$ (<full column rank>), whereas <surjectivity> means <matrix rank> $m$ (<full row rank>). A rectangular <matrix> can satisfy one condition without the other; its <kernel> and <image of a linear map> give the useful tests without calling it a <singular matrix>.

Exact singularity also differs from sensitivity. For $0<|t|\leq1$, $A(t)$ is an <invertible matrix>, but its smallest <singular value> is $|t|$ and its <condition number> in <Euclidean norm> is $1/|t|$. A change $\delta b=(0,\varepsilon)$ changes the recovered input by $\delta x=(0,\varepsilon/t)$: with $t=1/100$ and $\varepsilon=1/1000$, an output change of $0.001$ causes an input change of $0.1$. These are exact toy values, not measurements. The <condition number> quantifies this sensitivity before a direction is lost; see https://www.cs.cornell.edu/courses/cs322/2007sp/notes/condition.pdf[Steve Marschner, Condition numbers, §2]. A small <determinant> alone is not a reliable sensitivity test: $\varepsilon I_2$ has <determinant> $\varepsilon^2$ yet <condition number> one for every nonzero real $\varepsilon$, since both <singular values> shrink equally.


 Singular matrix (source code)