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- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/02%3A_Matrices/2.05%3A_The_TransposeAnother important operation on matrices is that of taking the transpose.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.03%3A_General_Notation_Transposes_and_InversesThe solution for A−1 is found to be A−1=1ad−bc(d−b−ca). The factor in front of the matrix is the definition...The solution for A−1 is found to be \boldsymbol{\text{A}^{-1}=\frac{1}{ad-bc}\left(\begin{array}{rr}d&-b\\-c&a\end{array}\right).\label{eq:4}} The factor in front of the matrix is the definition of the determinant for our two-by-two matrix A: det The determinant of a two-by-two matrix is the product of the diagonals minus the product of the off-diagonals.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_Determinants/3.2E%3A_Exercises_for_Section_3.2This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and or...This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and orthogonal matrices, and provides proofs regarding matrix similarities that maintain determinant values.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_DeterminantsThere are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of...There are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of row operations on the determinant of a matrix. In future sections, we will see that using the following properties can greatly assist in finding determinants. This section will use the theorems as motivation to provide various examples of the usefulness of the properties.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/12%3A_Matrices_and_Determinants/12.05%3A_The_TransposeAnother important operation on matrices is that of taking the transpose.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/12%3A_Supplementary_notes_on_matrices_and_linear_systems/12.05%3A_Special_operations_on_matricesGiven positive integers m, n \in \mathbb{Z}_+ and any matrix A = (a_{ij} ) \in \mathbb{F}^{m \times n} , we define the transpose A^T = ((a^T )_{ij} ) \in \mathbb{F}^{n \times m} and the con...Given positive integers m, n \in \mathbb{Z}_+ and any matrix A = (a_{ij} ) \in \mathbb{F}^{m \times n} , we define the transpose A^T = ((a^T )_{ij} ) \in \mathbb{F}^{n \times m} and the conjugate transpose A^{\ast} = ((a^{\ast} )_{ij} ) \in \mathbb{F}^{n \times m} by is orthogonal if A \in GL(n, \mathbb{R}) and A^{-1} = A^T . Moreover, we define the (real) orthogonal group to be the set O(n) = \{A \in GL(n, \mathbb{R})~ |~ A^{-1} = A^T \}.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/07%3A_Matrices/7.03%3A_Properties_of_MatricesThe objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of ef...The objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of efficient computation.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/12%3A_Matrices/12.02%3A_Properties_of_MatricesThe objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of ef...The objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of efficient computation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/03%3A_Operations_on_Matrices/3.01%3A_The_Matrix_TransposeThe transpose of a matrix is an operator that flips a matrix over its diagonal. Transposing a matrix essentially switches the row and column indices of the matrix.
- https://math.libretexts.org/Workbench/Math_26%3A_Introduction_to_Linear_Algebra/02%3A_Linear_Transformations_and_Matrix_Algebra/2.05%3A_Determinants-_DefinitionIn this section, we define the determinant, and we present one way to compute it. Then we discuss some of the many wonderful properties the determinant enjoys.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/02%3A_Matrices/2.05%3A_The_TransposeAnother important operation on matrices is that of taking the transpose.
