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This shows you the differences between two versions of the page.
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linear_algebra:04_invertible_matrices [2014/01/13 20:52] marje created |
linear_algebra:04_invertible_matrices [2014/01/20 13:39] marje |
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| </box> | </box> | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: Verify the lemma in case of $A=\begin{pmatrix}1&3&2&4\\-4&-2&-3&-1\\5&8&6&7\end{pmatrix}$ by calculating | + | Verify the lemma in case of $A=\begin{pmatrix}1&3&2&4\\-4&-2&-3&-1\\5&8&6&7\end{pmatrix}$ by calculating |
| ;#; | ;#; | ||
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| means just calculating the product $Db$. | means just calculating the product $Db$. | ||
| - | <WRAP indent> | + | <WRAP> |
| - | :?: Let $A=\begin{pmatrix}1&0&1\\1&-1&1\\1&1&-1\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}1\\2\\3\end{pmatrix}$, | + | Let $A=\begin{pmatrix}1&0&1\\1&-1&1\\1&1&-1\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}1\\2\\3\end{pmatrix}$, |
| and $D=\begin{pmatrix}0&\frac{1}{2}&\frac{1}{2}\\1&-1&0\\1&-\frac{1}{2}&-\frac{1}{2}\end{pmatrix}$.\\ | and $D=\begin{pmatrix}0&\frac{1}{2}&\frac{1}{2}\\1&-1&0\\1&-\frac{1}{2}&-\frac{1}{2}\end{pmatrix}$.\\ | ||
| Verify that $DA=I$ and solve the system $A\mathbf{x}=\mathbf{b}$ by usind the matrix $D$.\\ | Verify that $DA=I$ and solve the system $A\mathbf{x}=\mathbf{b}$ by usind the matrix $D$.\\ | ||
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| we can solve the system easily for many different values of $\mathbf{b}$. | we can solve the system easily for many different values of $\mathbf{b}$. | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: | + | |
| Solve the system of linear equations form previous exercise for | Solve the system of linear equations form previous exercise for | ||
| $\mathbf{b}=\begin{pmatrix}0\\1\\3\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}1\\1\\3\end{pmatrix}$. | $\mathbf{b}=\begin{pmatrix}0\\1\\3\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}1\\1\\3\end{pmatrix}$. | ||
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| Thus, for solving the system $A\mathbf{x}=\mathbf{b}$, we are looking for a $n\times n$ matrix $D$ | Thus, for solving the system $A\mathbf{x}=\mathbf{b}$, we are looking for a $n\times n$ matrix $D$ | ||
| such that $AD=I$. It is a matrix equation where $D$ is the unknown. From previous we know | such that $AD=I$. It is a matrix equation where $D$ is the unknown. From previous we know | ||
| - | (see [[linear_algebra:gaussian_elimination|Gaussian elimination algorithm]]) how to solved | + | (see [[linear_algebra:03_gaussian_elimination|Gaussian elimination algorithm]]) how to solved |
| similar equations where the unknown is and $n\times 1$ vector and on the right hand side is also an vector | similar equations where the unknown is and $n\times 1$ vector and on the right hand side is also an vector | ||
| - this is were we are heading. | - this is were we are heading. | ||
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| \qquad (\ast)$$ | \qquad (\ast)$$ | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: | + | |
| Calculate the second column of the product | Calculate the second column of the product | ||
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| $$ | $$ | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: | + | |
| Perform the above mentioned elementary row operations on the matrix | Perform the above mentioned elementary row operations on the matrix | ||
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| system even tough $A^{-1}$ is not calculated. | system even tough $A^{-1}$ is not calculated. | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: | + | |
| Find the inverse of the matrix | Find the inverse of the matrix | ||
| \begin{align*} | \begin{align*} | ||
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| </WRAP> | </WRAP> | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: | + | |
| Modify your IPython script for simple Gaussian elimination algorithm for finding the inverse of a given matrix. | Modify your IPython script for simple Gaussian elimination algorithm for finding the inverse of a given matrix. | ||
| </WRAP> | </WRAP> | ||