40.3: Invertible Systems
- Page ID
- 70543
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An invertible system has a square \(A\) that is invertible such that all the following properties are true:
- \( A^{-1}A = AA^{-1} = I \)
- \((A^{-1})^{-1} = A\)
- \((cA)^{-1} = \frac{1}{c}A^{-1}\)
- \((AB)^{-1} = B^{-1}A^{-1}\)
- \((A^n)^{-1} = (A^{-1})^n\)
- \((A^\top)^{-1} = (A^{-1})^\top\) here \(A^\top\) is the transpose of the matrix \(A\).
Consider the following system of equations:
\[\begin{split}\begin{bmatrix}5&-2&2 \\ 4 & -3 &4 \\ 4& -6 &7 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix}\end{split} \nonumber \]
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 \(x=A^{-1}b\)
Pick at least two methods to solve the system of equations and compare them.