40.3: Invertible Systems

An invertible system has a square \(A\) that is invertible such that all the following properties are true:

1. \( A^{-1}A = AA^{-1} = I \)
2. \((A^{-1})^{-1} = A\)
3. \((cA)^{-1} = \frac{1}{c}A^{-1}\)
4. \((AB)^{-1} = B^{-1}A^{-1}\)
5. \((A^n)^{-1} = (A^{-1})^n\)
6. \((A^\top)^{-1} = (A^{-1})^\top\) here \(A^\top\) is the transpose of the matrix \(A\).

Consider the following system of equations:

\[\begin{split}\begin{bmatrix}5&-2&2 \\ 4 & -3 &4 \\ 4& -6 &7 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix}\end{split} \nonumber \]

import numpy as np
import sympy as sym

A = np.matrix([[5, -2, 2], [4, -3, 4], [4,-6,7]])
b = np.matrix([[1],[2],[3]])
display(sym.Matrix(A))
display(sym.Matrix(b))

Iterative algorithms (Gauss-Seidel and Jacobian):

• They may require many iterations
• Gauss-Seidel is faster than Jacobian
• They do not work for all square invertible systems.

Non-iterative algorithms:

• Gauss elimination (rref)
• LU decomposition
• Find the inverse of the matrix \(A\) and \(x=A^{-1}b\)

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