05_inverses.md

Linear System of Equations

$$\begin{aligned} 2x + 5y + 3z = -3\\ 4x + 0y + 8z = 0\\ 1x + 3y + 0z = 2 \end{aligned} \Rightarrow \begin{bmatrix} 2 & 5 & 3\\ 4 & 0 & 8\\ 1 & 3 & 0 \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} -3\\ 0\\ 2 \end{bmatrix}$$

• We can then let $A = \begin{bmatrix} 2 & 5 & 3\ 4 & 0 & 8\ 1 & 3 & 0 \end{bmatrix}, \overrightarrow{x} = \begin{bmatrix} x\ y\ z \end{bmatrix},$ and $\overrightarrow{v} = \begin{bmatrix} -3\ 0\ 2 \end{bmatrix}$

Inverse

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

• $A^{-1}$ exists if and only if $\det A \ne 0$
• When determinant is 0, the linear transformation $\mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$.