Inverse matrices

$$A \in \mathbb{R}^{n\times n}$$ is invertible or nonsingular if $$\det A \neq 0$$, which is equivalent 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

Some identities for matrix inversion

 * Rank one update formula (known as Sherman-Morrison formula): Iterative inversions of the form $$P(m+1)^{-1} = \left(P(m) + a_{m+1}\right)^{-1}$$ can be done efficiently via:

$$(P+aa^T)^{-1} = P^{-1} - \frac{1}{1+a^T P^{-1}a} (P^{-1}a)(P^{-1}a)^T$$
 * $$(I-C)^{-1} = I+C+C^2+C^3+\dots$$ (if the series converges ). This is also called Neumann series.
 * Matrix inversion lemma (known as Woodbury matrix identity): $$(A+BC)^{-1} = A^{-1}-A^{-1} B (I+C A^{-1}B)^{-1}CA^{-1}$$, which is useful when $$A$$ is easy to invert and $$BC$$ is a low-rank matrix