7.3 Matrix inversion

The inverse of a square matrix $A \in \mathbb{R}^n$ is $A^{-1}$ . It is the unique matrix such that:

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

$A^{-1}$ is only defined if $\mathrm{det}(A)\neq 0$ , that is if $A$ is not singular.

Some properties:

\begin{align}(A^{-1})^{-1}&=A \\ (AB)^{-1}&=B^{-1}A^{-1} \\ (A^{-1})^\top &= (A^\top)^{-1}\triangleq A^{-\top} \end{align}

For a simple 2x2 matrix, we have:

A=\begin{bmatrix}a & b \\c &d\end{bmatrix},\;\;A^{-1}=\frac{1}{|A|}\begin{bmatrix}d & -b \\-c &a\end{bmatrix}

For a block diagonal matrix, we invert each block separately:

\begin{bmatrix}A & 0 \\0 &B\end{bmatrix}^{-1}=\begin{bmatrix}A^{—1} & 0 \\0 &B^{—1}\end{bmatrix}