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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/11%3A_Appendices/03%3A_Determinants_and_Cramer's_Rule_for_2_X_2_Matrices\[x_1= \frac{\det \left(A_{1}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} b_1 & a_{12} \\ b_2 & a_{22} \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_...\[x_1= \frac{\det \left(A_{1}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} b_1 & a_{12} \\ b_2 & a_{22} \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber\] and \[x_2= \frac{\det \left(A_{2}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} a_{11} & b_1 \\ a_{21} & b_2 \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber\]
- https://math.libretexts.org/Courses/Mt._San_Jacinto_College/Differential_Equations_(No_Linear_Algebra_Required)/08%3A_Appendices/8.03%3A_Determinants_and_Cramer's_Rule_for_2_X_2_Matrices\[x_1= \frac{\det \left(A_{1}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} b_1 & a_{12} \\ b_2 & a_{22} \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_...\[x_1= \frac{\det \left(A_{1}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} b_1 & a_{12} \\ b_2 & a_{22} \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber\] and \[x_2= \frac{\det \left(A_{2}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} a_{11} & b_1 \\ a_{21} & b_2 \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber\]
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.01%3A_Basic_TechniquesLet \(A\) be a square matrix. The determinant of \(A\), denoted by \( \det (A) \), is an important number that gives us some very useful information about the matrix. We will explore the determinant t...Let \(A\) be a square matrix. The determinant of \(A\), denoted by \( \det (A) \), is an important number that gives us some very useful information about the matrix. We will explore the determinant throughout this section and chapter.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/03%3A_Determinants/3.01%3A_Basic_TechniquesLet A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/03%3A_Determinants/3.01%3A_Basic_TechniquesLet A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/03%3A_Determinants/3.01%3A_Basic_TechniquesLet A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/03%3A_Determinants/3.01%3A_Basic_TechniquesLet A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
- 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/Courses/Ohio_Northern_University/Differential_Equations_and_Linear_Algebra_(Anup_Lamichhane)/06%3A_Eigenvalues_and_Eigenvectors/6.02%3A_Properties_of_Eigenvalues_and_EigenvectorsThis page discusses eigenvalues and eigenvectors of matrices, highlighting their relationships, particularly between a matrix, its transpose, and its inverse. It explains that \(A\) and \(A^T\) share ...This page discusses eigenvalues and eigenvectors of matrices, highlighting their relationships, particularly between a matrix, its transpose, and its inverse. It explains that \(A\) and \(A^T\) share eigenvalues, and presents the Eigenvalue Theorem related to matrix invertibility. Examples illustrate eigenvalue calculations, including cases with repeated and complex eigenvalues, and underscore the significance of zero eigenvalues.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/12%3A_Matrices_and_Determinants/12.08%3A_Basic_Techniques_of_DeterminantsLet A be an n×n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a very important number which we will explore throughout this section.
- https://math.libretexts.org/Courses/Canada_College/Linear_Algebra_and_Its_Application/03%3A_Determinants/3.01%3A_Basic_TechniquesLet \(A\) be a square matrix. The determinant of \(A\), denoted by \( \det (A) \), is an important number that gives us some very useful information about the matrix. We will explore the determinant t...Let \(A\) be a square matrix. The determinant of \(A\), denoted by \( \det (A) \), is an important number that gives us some very useful information about the matrix. We will explore the determinant throughout this section and chapter.
