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4: Unary Matrix Operations
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.04%3A_Unary_Matrix_Operations
\[\begin{split} \left\lbrack A \right\rbrack &= \left\lbrack $\begin{matrix} 25 \\ 5 \\ 6 \\ \end{matrix}$ $\begin{matrix} 20 \\ 10 \\ 16 \\ \end{matrix}$ $\begin{matrix} 3 \\ 15 \\ 7 \\ \end{matrix}$ \begin{m...\[\begin{split} \left\lbrack A \right\rbrack &= \left\lbrack $\begin{matrix} 25 \\ 5 \\ 6 \\ \end{matrix}$ $\begin{matrix} 20 \\ 10 \\ 16 \\ \end{matrix}$ $\begin{matrix} 3 \\ 15 \\ 7 \\ \end{matrix}$ $\begin{matrix} 2 \\ 25 \\ 27 \\ \end{matrix}$ \right\rbrack\\ &= \left\lbrack $\begin{matrix} 33.25 \\ 40.19 \\ 25.03 \\ \end{matrix}$ $\begin{matrix} 30.01 \\ 38.02 \\ 22.02 \\ \end{matrix}$ $\begin{matrix} 35.02 \\ 41.03 \\ 27.03 \\ \end{matrix}$ $\begin{matrix} 30.05 \\ 38.23 \\ 22.95 \\ \end{matrix}$ \right\rbrack…
5: System of Equations
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.05%3A_System_of_Equations
\[\begin{bmatrix} a_{11} & a_{12} & \cdot & \cdot & a_{1n} \\ a_{21} & a_{22} & \cdot & \cdot & a_{2n} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ a_{n1} & a_{... $\begin{bmatrix} a_{11} & a_{12} & \cdot & \cdot & a_{1n} \\ a_{21} & a_{22} & \cdot & \cdot & a_{2n} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ a_{n1} & a_{n2} & \cdot & \cdot & a_{nn} \\ \end{bmatrix}\begin{bmatrix} a_{11}^{'} \\ a_{21}^{'} \\ \cdot \\ \cdot \\ a_{n1}^{'} \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ \cdot \\ \cdot \\ 0 \\ \end{bmatrix} \nonumber$
2: Vectors
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.02%3A_Vectors
If we were given another vector ${\overrightarrow{A}}_{4}$ , the rank of the set of the vectors \({\overrightarrow{A}}_{1},\ {\overrightarrow{A}}_{2},\ {\overrightarrow{A}}_{3},{\overrightarrow{A}}_{...If we were given another vector ${\overrightarrow{A}}_{4}$ , the rank of the set of the vectors ${\overrightarrow{A}}_{1},\ {\overrightarrow{A}}_{2},\ {\overrightarrow{A}}_{3},{\overrightarrow{A}}_{4}$ would still be 3 as the rank of a set of vectors is always less than or equal to the dimension of the vectors and that at least ${\overrightarrow{A}}_{1},\ {\overrightarrow{A}}_{2},\ {\overrightarrow{A}}_{3}$ are linearly independent.
InfoPage
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/00%3A_Front_Matter/02%3A_InfoPage
The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californ...The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
Index
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/zz%3A_Back_Matter/10%3A_Index
6: Gaussian Elimination Method for Solving Simultaneous Linear Equations
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.06%3A_Gaussian_Elimination_Method_for_Solving_Simultaneous_Linear_Equations
\[\begin{split} &a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + \ldots + a_{1n}x_{n} = b_{1}\\ &{a^\prime }_{22}x_{2} + {a^\prime }_{23}x_{3} + \ldots + {a^\prime }_{2n}x_{n} = {b^\prime }_{2}\\ &{a^\prime...\[\begin{split} &a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + \ldots + a_{1n}x_{n} = b_{1}\\ &{a^\prime }_{22}x_{2} + {a^\prime }_{23}x_{3} + \ldots + {a^\prime }_{2n}x_{n} = {b^\prime }_{2}\\ &{a^\prime }_{32}x_{2} + {a^\prime }_{33}x_{3} + \ldots + {a^\prime }_{3n}x_{n} = {b^\prime }_{3}\\ &\vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\\ &{a^\prime }_{n2}x_{2} + {a^\prime }_{n3}x_{3} + \ldots + {a^\prime }_{\text{nn}}x_{n} = {b^\prime }_{n} \end…
10: Eigenvalues and Eigenvectors
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.10%3A_Eigenvalues_and_Eigenvectors
\[\begin{split} $\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix}$ &= $\begin{bmatrix} - a \\ b \\ a \\ \end{bmatrix}$ \\ & = $\begin{bmatrix} - a \\ 0 \\ a \\ \end{bmatrix}$ + \begin{bmatrix} 0 \\ b... $\begin{split} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} &= \begin{bmatrix} - a \\ b \\ a \\ \end{bmatrix}\\ & = \begin{bmatrix} - a \\ 0 \\ a \\ \end{bmatrix} + \begin{bmatrix} 0 \\ b \\ 0 \\ \end{bmatrix}\\ &= a\begin{bmatrix} - 1 \\ 0 \\ 1 \\ \end{bmatrix} + b\begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix} \end{split} \nonumber$
1: Introduction
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.01%3A_Introduction
\[\lbrack A\rbrack = $\begin{bmatrix} a_{11} & a_{12} & {.......} & a_{1n} \\ a_{21} & a_{22} & {.......} & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & {.......} & a_{mn} \\ \end{bmatrix}$ \nonumb... $\lbrack A\rbrack = \begin{bmatrix} a_{11} & a_{12} & {.......} & a_{1n} \\ a_{21} & a_{22} & {.......} & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & {.......} & a_{mn} \\ \end{bmatrix}\nonumber$ The answer is simple – the definition of a weak(ly) diagonally dominant matrix is identical to that of a diagonally dominant matrix as the inequality used for the check is a weak inequality of greater than or equal to ( $\geq$ ).
TitlePage
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/00%3A_Front_Matter/01%3A_TitlePage
Introduction to Matrix Algebra Autar Kaw
Chapters
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters
3: Binary Matrix Operations
https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.03%3A_Binary_Matrix_Operations
\[\begin{split} \left\lbrack C \right\rbrack &= \left\lbrack A \right\rbrack + \left\lbrack B \right\rbrack\\ &= \begin{bmatrix} $\begin{matrix} 25 & 20 \\ \end{matrix}$ & \begin{matrix} 3 & 2 \\ \end{m...\[\begin{split} \left\lbrack C \right\rbrack &= \left\lbrack A \right\rbrack + \left\lbrack B \right\rbrack\\ &= $\begin{bmatrix} \begin{matrix} 25 & 20 \\ \end{matrix} & \begin{matrix} 3 & 2 \\ \end{matrix} \\ \begin{matrix} 5 & 10 \\ \end{matrix} & \begin{matrix} 15 & 25 \\ \end{matrix} \\ \begin{matrix} 6 & 16 \\ \end{matrix} & \begin{matrix} 7 & 27 \\ \end{matrix} \\ \end{bmatrix}$ + \begin{bmatrix} $\begin{matrix} 20 & 5 \\ \end{matrix}$ & $\begin{matrix} 4 & 0 \\ \end{matrix}$ \\ \begin{matrix}…

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