Inverse matrix

An inverse matrix, or inverse of a matrix, is a non-singular square matrix that produces the identity matrix when multiplied by or to its corresponding inverse. For a given matrix $A\in {\mathcal {M}}_{n\times n}$, the inverse exists if and only if the determinant of $A$ is non-zero, $\det A\neq 0$. Such a matrix is called a non-singular matrix. The inversion of a matrix is itself invertible, i.e. $(A^{-1})^{-1}=A$

The formula for finding the inverse of a 2x2 matrix is as follows:

$A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\Rightarrow A^{-1}={\frac {1}{ad-bc}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}$.

However, this formula does not hold for matrices larger than 2x2.