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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/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/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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.03%3A_Application_of_the_Determinant_to_Inverses_Cramer's_Rule/3.3E%3A_Exercises_for_Section_3.3This page outlines exercises on matrix operations, including determining invertibility via determinants, calculating adjugates and inverses, and using Cramer’s Rule for solving equations. It reveals t...This page outlines exercises on matrix operations, including determining invertibility via determinants, calculating adjugates and inverses, and using Cramer’s Rule for solving equations. It reveals that matrix A is invertible, while matrix B is not, highlighting the role of determinants in unique solutions. Practical applications in electrical circuits are discussed along with challenges addressing numerical stability and conditions for unique solutions.
- 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/Irvine_Valley_College/Math_26%3A_Introduction_to_Linear_Algebra/02%3A_Linear_Transformations_and_Matrix_Algebra/2.06%3A_Cofactor_ExpansionsRecipes: the determinant of a \(n\times n\) matrix for \(n\ge 3\), compute the determinant using cofactor expansions. In this section, we give a recursive formula for the determinant of a matrix, call...Recipes: the determinant of a \(n\times n\) matrix for \(n\ge 3\), compute the determinant using cofactor expansions. In this section, we give a recursive formula for the determinant of a matrix, called a cofactor expansion. The formula is recursive in that we will compute the determinant of an \(n\times n\) matrix assuming we already know how to compute the determinant of an \((n-1)\times(n-1)\) matrix.