Inverse

Definition: A square matrix $A$ is invertible (or nonsingular) if there exists a matrix $A^{-1}$ such that:

$$AA^{-1} = A^{-1}A = I$$

where $I$ is the identity matrix.

Example:
$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$:
$$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Properties:

• $(A^{-1})^{-1} = A$
• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^T)^{-1} = (A^{-1})^T$
• $\det(A^{-1}) = 1/\det(A)$

A matrix is invertible iff:

• $\det(A) \neq 0$
• Rank of $A$ equals $n$
• Rows/columns are linearly independent
• 0 is not an eigenvalue