Inverse Matrix

Definition

${\mathbf{A}}^{-1}$ satisfies $\mathbf{A}{\mathbf{A}}^{-1}={\mathbf{A}}^{-1}\mathbf{A}=\mathbf{I}$ for given matrix $\mathbf{A}$

Computation

2 x 2 matrix

${\mathbf{A}}^{-1}=\frac{1}{|\mathbf{A}|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ where the matrix $\mathbf{A}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, and $|\mathbf{A}|$ is a Determinant of the matrix

Using Cofactor

$\mathbf{A}\mathrm{adj}(\mathbf{A})={\mathbf{AC}}^{⊺}=\text{det}(\mathbf{A})\mathbf{I}$ where the matrix $\mathbf{C}=\left[\begin{array}{cccc}{C}_{11}& C_{12}& \cdots & C_{1n}\\ C_{21}& C_{22}& \cdots & C_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ C_{n1}& C_{n2}& \cdots & C_{nn}\end{array}\right]$ is the cofactor matrix, which is formed by all the cofactors of a given matrix $\mathbf{A}$. Then, by the definition of inverse matrix, ${\mathbf{A}}^{-1}=\frac{1}{\text{det}(\mathbf{A})}{\mathbf{C}}^{⊺}$