[Linear Algebra] Inverse Matrix

Inverse Matrix

• Definition : For a square matrix $A \in \mathbb{R}^{n \times n}$, its inverse matrix $A^{-1}$ is defined such that $A^{-1}A = A A^{-1} = I_n$

Solving Linear System via Inverse Matrix

• We can now solve $A \textbf{x} = \textbf{b}$ as follows

$A \textbf{x} = \textbf{b}$
$A^{-1} A \textbf{x} = A^{-1} \textbf{b}$
$I_n \textbf{x} = A^{-1} \textbf{b}$
$\textbf{x} = A^{-1} \textbf{b}$

e.g.

$A \textbf{x} = \textbf{b}$

where $A = \begin{bmatrix} 2 & 2 & 1 \\ 3 & 1 & 0 \\ 1 & 3 & 1 \\ \end{bmatrix}$, $\textbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$, $\textbf{b} = \begin{bmatrix} 6 \\ 14 \\ 18 \end{bmatrix}$

$A^{-1} = \begin{bmatrix} 0.25 & 0.25 & -0.25 \\ -0.75 & 0.25 & 0.75 \\ 2 & -1 & -1 \\ \end{bmatrix}$

$\textbf{x} = A^{-1} \textbf{b} = \begin{bmatrix} 0.25 & 0.25 & -0.25 \\ -0.75 & 0.25 & 0.75 \\ 2 & -1 & -1 \\ \end{bmatrix}\begin{bmatrix} 6 \\ 14 \\ 18 \end{bmatrix} = \begin{bmatrix} 0.5 \\ 12.5 \\ -5 \end{bmatrix}$