Nonsingular matrix

An $$n\times n$$ matrix $$A$$ is nonsingular (or invertible) if its rank is $$n$$. Another very typical characteristic of a nonsingular matrix is that $$\det A \neq 0$$. A nonsingular matrix is always a product of elementary matrices. [See (3.9.3) in ]

$$A \in \mathbb{R}^{n\times n}$$ is invertible or nonsingular is equivalent also to the following conditions:
 * Columns of $$A$$ are a basis for $$\mathbb{R}^n$$
 * Rows of $$A$$ are also a basis for $$\mathbb{R}^n$$
 * $$y=Ax$$ has a unique solution $$x$$ for every $$y \in \mathbb{R}^n$$
 * A has a (left and right) inverse denoted $$A^{-1} \in \mathbb{R}^{n\times n}$$ with $$AA^{-1} = A^{-1}A = I$$
 * $$\mathcal{N}(A) = \{0\}$$
 * $$\mathcal{R}(A) = \mathbb{R}^n$$
 * $$\det A^TA=\det AA^T \neq 0$$
 * $$A$$ does not have a zero eigenvalue