Chapter 13 Linear Equations and Matrices

Enterprise 2023-09-30 12:50:22 views: null

13.1 Examples of systems of linear equations

Suppose a system of linear equations is:

$\begin{cases}2x-5y=7\\-2x+4y=-6\end{cases}$

Converted to matrix form:

$\begin{bmatrix}2&-5\\-2&4\end{bmatrix}\;\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\-6\end{bmatrix}\$

Assumptions:

$A=\begin{bmatrix}2&-5\\-2&4\end{bmatrix}\;\vec{x}=\begin{bmatrix}x\\y\end{bmatrix}\;\vec{b}=\begin{bmatrix}7\\-6\end{bmatrix}\$

The above equation becomes:

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

$A^{-1}A\vec{x}=A^{-1}b$

$\vec{x}=A^{-1}b$

Assumptions:

$A = \begin{bmatrix}a&b\\c&d\end{bmatrix}\$

Its inverse matrix is:

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