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11.3: Determinants and Cramer's Rule for 2 X 2 Matrices
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 $x_2= \frac{\det \left(A_{2}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} a_{11} & b_1 \\ a_{21} & b_2 \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber$
4.3: Properties of the Determinant
https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/04%3A_Determinants/4.03%3A_Properties_of_the_Determinant
The determinant, as we know, is a function that maps an n -by- n matrix to a scalar. We now define this determinant function by the following three properties.
1.5: The Cross Product
https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/01%3A_Vectors_in_Space/1.05%3A_The_Cross_Product
In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, a...In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, and we examine torque in more detail later in the section.
3.2: Properties of Determinants
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_Determinants
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...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.
3.2E: Exercises for Section 3.2
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_3.2
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 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.
6.4E: Exercises for Section 6.4
https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.04%3A_Matrices_and_linear_systems/6.4E%3A_Exercises_for_Section_6.4
This page features an exercise set focused on matrix operations, including solving systems of equations with matrix inverses, computing determinants, and exploring matrix invertibility conditions. It ...This page features an exercise set focused on matrix operations, including solving systems of equations with matrix inverses, computing determinants, and exploring matrix invertibility conditions. It includes tasks like determining invertibility for matrices with variables, verifying matrix relationships, and provides solutions for all exercises. Key examples focus on determinant calculation and identifying nonzero matrices that meet specific multiplication criteria.
1.4: Cross Product
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/01%3A_Vectors_in_Euclidean_Space/1.04%3A_Cross_Product
In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vect...In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector. This product, called the cross product, is only defined for vectors in $\mathbb{R}^{3}$ . The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly.
1.5: The Cross Product
https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/01%3A_Vectors_in_Space/1.05%3A_The_Cross_Product
In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, a...In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, and we examine torque in more detail later in the section.
11.6: A.6- Determinant
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/11%3A_Appendix_A-_Linear_Algebra/11.06%3A_A.6-_Determinant
For example, if \[\text{If} \qquad A = $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ , \qquad \text{then} \qquad A_{12} = $\begin{bmatrix} 4 & 6 \\ 7 & 9 \end{bmatrix}$ \qquad \text{...For example, if $\text{If} \qquad A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} , \qquad \text{then} \qquad A_{12} = \begin{bmatrix} 4 & 6 \\ 7 & 9 \end{bmatrix} \qquad \text{and} \qquad A_{23} = \begin{bmatrix} 1 & 2 \\ 7 & 8 \end{bmatrix} . \nonumber$ We now define the determinant recursively $\det (A) \overset{\text{def}}{=} \sum_{j=1}^n {(-1)}^{1+j} a_{1j} \det (A_{1j}) , \nonumber$ or in other words \[\det (A) = a_{11} \det (A_{11}) - a_{12} \det (A_{12}) + a_…
A.6: Determinant
https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/Appendix_A%3A_Linear_Algebra/A.6%3A_Determinant
The page explains the concept of the determinant of square matrices, including definitions for $1 \times 1$ and $2 \times 2$ matrices. It elaborates on how determinants map $n$ -dimensional space...The page explains the concept of the determinant of square matrices, including definitions for $1 \times 1$ and $2 \times 2$ matrices. It elaborates on how determinants map $n$ -dimensional spaces and change the area of objects, explaining the determinant as a factor indicating changes in area and orientation. The cofactor expansion is introduced for computing determinants of larger matrices.
8.3: Determinants and Cramer's Rule for 2 X 2 Matrices
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 $x_2= \frac{\det \left(A_{2}\right)}{\det \left(A\right)} = \frac{\left| \begin{array}{rrr} a_{11} & b_1 \\ a_{21} & b_2 \end{array} \right| }{\left| \begin{array}{rrr} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| }\nonumber$

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