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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] (current)marje 2014/01/20 13:39 marje 2014/01/13 20:52 marje created 2014/01/20 13:39 marje 2014/01/13 20:52 marje created Line 18: Line 18: ​ - <​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 ;#; ;#; Line 46: Line 46: 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 $D$.\\ Line 56: Line 56: 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 $\mathbf{b}=\begin{pmatrix}1\\1\\3\end{pmatrix}$. Line 74: Line 73: 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. Line 94: Line 93: \qquad (\ast)$$\qquad (\ast)$$ - <​WRAP ​indent> + <​WRAP ​task> - :?: + Calculate the second column of the product Calculate the second column of the product Line 245: Line 243:  - <​WRAP ​indent> + <​WRAP ​task> - :?: + Perform the above mentioned elementary row operations on the matrix Perform the above mentioned elementary row operations on the matrix Line 280: Line 277: 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*} Line 294: Line 290: ​ - <​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. ​ 