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Title: Determinant - Wikipedia

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.
The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.
The determinant of a 2 × 2 matrix is
and the determinant of a 3 × 3 matrix is
The determinant of an n × n matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of
n
!
{\displaystyle n!}
(the factorial of n) signed products of matrix entries. It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant, equal to the product of the diagonal entries of the row echelon form.
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties:
The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism of a finite-dimensional vector space, the determinant of the matrix that represents it on a basis does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depend on the choice of a coordinate system.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the n-dimensional volume are transformed under the endomorphism. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
The determinant of a 2 × 2 matrix
(
a
b
c
d
)
{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
is denoted either by "det" or by vertical bars around the matrix, and is defined as
For example,
The determinant has several key properties that can be proved by direct evaluation of the definition for
2
×
2
{\displaystyle 2\times 2}
-matrices, and that continue to hold for determinants of larger matrices. They are as follows:[1] first, the determinant of the identity matrix
(
1
0
0
1
)
{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
is 1.
Second, the determinant is zero if two rows are the same:
This holds similarly if the two columns are the same. Moreover,
Finally, if any column is multiplied by some number
r
{\displaystyle r}
(i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:

Source: [[0]](https://en.wikipedia.org/wiki/Determinant)


Title: Invertible matrix - Wikipedia

In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. Invertible matrices are the same size as their inverse.
An n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or rarely regular) if there exists an n-by-n square matrix B such that
A
B
=
B
A
=
I
n
,
{\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},}
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.[1] If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.[2]
Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = In. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im.
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group, the general linear group of degree n, denoted GLn(R).
Let A be a square n-by-n matrix over a field K (e.g., the field ⁠
R
{\displaystyle \mathbb {R} }
⁠ of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:[3]
Furthermore, the following properties hold for an invertible matrix A:
The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where the rows of V are denoted as
v
i
T
{\displaystyle v_{i}^{\mathrm {T} }}
and the columns of U as
u
j
{\displaystyle u_{j}}
for
1
≤
i
,
j
≤
n
.
{\displaystyle 1\leq i,j\leq n.}
Then clearly, the Euclidean inner product of any two
v
i
T
u
j
=
δ
i
,
j
.
{\displaystyle v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.}

Source: [[1]](https://en.wikipedia.org/wiki/Invertible_matrix)


Title: Cross product - Wikipedia

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here
E
{\displaystyle E}
), and is denoted by the symbol
×
{\displaystyle \times }
. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b,[1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product is zero.[2]
The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition, that is, a × (b + c) = a × b + a × c.[1] The space
E
{\displaystyle E}
together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it is why an oriented space is needed). The resultant vector is invariant of rotation of basis. Due to the dependence on handedness, the cross product is said to be a pseudovector.
In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space.
The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product; one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[3] The cross-product in seven dimensions has undesirable properties, however (e.g. it fails to satisfy the Jacobi identity), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time.[4] (See § Generalizations below for other dimensions.)
The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics and applied mathematics, the wedge notation a ∧ b is often used (in conjunction with the name vector product),[5][6][7] although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to n dimensions.
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule[1] and a magnitude equal to the area of the parallelogram that the vectors span.[2]
The cross product is defined by the formula[8][9]
where
If the vectors a and b are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

Source: [[2]](https://en.wikipedia.org/wiki/Cross_product)


Title: C Program to Perform Matrix Division | CodeToFun

Photo Credit to CodeToFun
Matrix operations are fundamental in linear algebra and various scientific and engineering applications. One essential matrix operation is division.
In this tutorial, we'll explore a C program that performs matrix division, providing a foundational understanding of how to handle matrices in a programming context.
Let's delve into the C code that performs matrix division.
To test the program with different matrices, replace the values of matrixA and matrixB in the main function.
Compile and run the program to see the result of matrix division.
Matrix division is not a straightforward operation like addition or multiplication.
Matrix division involves multiplying one matrix by the inverse of another. Ensure that the second matrix is invertible, and additional logic for inverse calculation may be required.
For two matrices A and B, the division A/B is equivalent to A * B^(-1), where B^(-1) is the inverse of matrix B.
This basic example assumes a 2x2 matrix for simplicity. For larger matrices, you may need to implement a robust algorithm for matrix inversion.
Feel free to incorporate and modify this code as needed for your specific use case. Happy coding!
For over eight years, I worked as a full-stack web developer. Now, I have chosen my profession as a full-time blogger at codetofun.com.
Buy me a coffee to make codetofun.com free for everyone.
If you have any doubts regarding this article (C Program to Perform Matrix Division), please comment here. I will help you immediately.
dont i have to find the inverse matrix for division?
To perform division in matrix operations, you need to find the inverse of the matrix. I’ll be happy to write a tutorial on how to calculate the inverse matrix for division in detail and will write sooner.

Source: [[3]](https://codetofun.com/c/matrix-division/)


Title: Adding Matrices Worksheet

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Source: [[4]](https://studyzonemandarins.z21.web.core.windows.net/adding-matrices-worksheet.html)


Instruction: Using the provided web search results, to write a comprehensive reply to the user request.
Make sure to add the sources of cites using [[Number]](Url) notation after the reference. Example: [[0]](http://google.com)

User request:
Let A is 3x3 matrix (1,4,7,8,3,0,1,9,2)

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