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
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