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Created page with "== Introduction == The matrix product, also known as matrix multiplication, is a fundamental operation in linear algebra. It involves multiplying two matrices by following a specific arithmetic and summation procedure. This operation is not only pivotal in mathematics but also in various applications like physics, computer science, and engineering. == Basics of Matrix Multiplication == === Definition === Matrix multiplication of two matrices, A and B, results in a new..."

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== Introduction ==
The matrix product, also known as matrix multiplication, is a fundamental operation in linear algebra. It involves multiplying two matrices by following a specific arithmetic and summation procedure. This operation is not only pivotal in mathematics but also in various applications like physics, computer science, and engineering.

== Basics of Matrix Multiplication ==

=== Definition ===
Matrix multiplication of two matrices, A and B, results in a new matrix C. The entry in the i-th row and j-th column of C, denoted as C[i, j], is calculated as the sum of the products of corresponding elements from the i-th row of A and the j-th column of B.

he formula for a generic term ''cij'' in the matrix product C, which results from multiplying two matrices A and B, is as follows:

''cij''=∑''k''=1->''n''''aik''×''bkj''

In this formula:

* ''cij'' is the element in the i-th row and j-th column of the resulting matrix C.
* ''aik'' is the element in the i-th row and k-th column of matrix A.
* ''bkj'' is the element in the k-th row and j-th column of matrix B.
* The summation runs over k from 1 to n, where n is the number of columns in A (which is also the number of rows in B, ensuring the matrices are compatible for multiplication).

=== Conditions for Multiplicability ===
Two matrices A and B can be multiplied if and only if the number of columns in A is equal to the number of rows in B.

== Step-by-Step Example ==

=== Matrices to Multiply ===
Consider two 4x4 matrices A and B:

==== Matrix A ====
{| class="wikitable"
!
!Col 1
!Col 2
!Col 3
!Col 4
|-
|Row 1
|''a''11
|''a''12
|''a''13
|''a''14
|-
|Row 2
|''a''21
|''a''22
|''a''23
|''a''24
|-
|Row 3
|''a''31
|''a''32
|''a''33
|''a''34
|-
|Row 4
|''a''41
|''a''42
|''a''43
|''a''44
|}

==== Matrix B ====
{| class="wikitable"
!
!Col 1
!Col 2
!Col 3
!Col 4
|-
|Row 1
|''b''11
|''b''12
|''b''13
|''b''14
|-
|Row 2
|''b''21
|''b''22
|''b''23
|''b''24
|-
|Row 3
|''b''31
|''b''32
|''b''33
|''b''34
|-
|Row 4
|''b''41
|''b''42
|''b''43
|''b''44
|}

=== Computation Steps ===

===== Generic =====
For the specific case of 4x4 matrices:

''cij''=''ai''1×''b''1''j''+''ai''2×''b''2''j''+''ai''3×''b''3''j''+''ai''4×''b''4''j''

This equation calculates each element of the resulting matrix by summing the products of corresponding elements from the i-th row of A and the j-th column of B.

===== Example =====

# Calculate C[1, 1]: ''C''[1,1]=''a''11×''b''11+''a''12×''b''21+''a''13×''b''31+''a''14×''b''41
# Calculate C[1, 2]: ''C''[1,2]=''a''11×''b''12+''a''12×''b''22+''a''13×''b''32+''a''14×''b''42
#* Continue this pattern across the first row of A and the first column of B.
# Calculate C[2, 1]: ''C''[2,1]=''a''21×''b''11+''a''22×''b''21+''a''23×''b''31+''a''24×''b''41
#* Follow the same method for the remaining entries of C.

=== Final Matrix C (Result of A x B) ===
{| class="wikitable"
!
!Col 1
!Col 2
!Col 3
!Col 4
|-
|Row 1
|''c''11
|''c''12
|''c''13
|''c''14
|-
|Row 2
|''c''21
|''c''22
|''c''23
|''c''24
|-
|Row 3
|''c''31
|''c''32
|''c''33
|''c''34
|-
|Row 4
|''c''41
|''c''42
|''c''43
|''c''44
|}
where each c_{ij} is computed using the above method.

== Conclusion ==
Matrix multiplication is a cornerstone of linear algebra and is essential in various scientific and mathematical applications. Understanding its computation process, provides a deeper insight into matrix operations and their significance in diverse fields.

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