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Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to produce a third. Unlike regular multiplication, it is not commutative, meaning that the order in which you multiply matters.
In mathematical terms, for two matrices A and B to be multiplied, the number of columns in A must be equal to the number of rows in B. The element in the resulting matrix at position (i, j) is calculated as the sum of the products of the corresponding elements of the i-th row of matrix A and the j-th column of matrix B.
Here is the general formula for the multiplication of two matrices:
If A is an m x n matrix and B is an n x p matrix, their product AB is an m x p matrix where each element is given by:
$$ (AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj} $$
Using this, let's multiply matrices A and B from the original problem: A = \[ \begin{array}{ll} a & b \ c & d \end{array} \], B = \[ \begin{array}{ll} 1 & 2 \ 2 & 1 \end{array} \].
Calculating AB:
$$ AB = \[ \begin{array}{ll} a & b \ c & d \end{array} \] \[ \begin{array}{ll} 1 & 2 \ 2 & 1 \end{array} \] = \[ \begin{array}{ll} a+2b & 2a+b \ c+2d & 2c+d \end{array} \] $$

Linear Algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
One of the essential tools in linear algebra is the matrix, especially when it comes to solving systems of linear equations, performing transformations, and optimizing operations. Mathematically, linear algebra deals with structures such as:

• Vectors and vector spaces
• Linear transformations
• Matrices and determinants
• Eigenvalues and eigenvectors

Considering our matrices A and B from the problem, we want to find conditions where the matrices commute, meaning that: $$ AB = BA $$
This is important in many applications, including solving systems of linear equations, computer graphics, and more. In this case, proving that two matrices commute involves performing matrix multiplication and equating the resulting matrices.
Solving these systematically leads us to a deeper understanding of linear independence, eigenvectors, and other critical concepts in linear algebra.

A system of linear equations is a collection of one or more linear equations involving the same set of variables. Such systems can be solved using various methods such as substitution, elimination, and matrix operations like Gaussian elimination.
In our problem, to determine the values of a, b, c, and d for which the matrices A and B commute, we created a system of equations by equating AB to BA and then comparing corresponding elements:
\[ \begin{array}{ll} a+2b = a+2c \ 2a+b = b+2d \ c+2d = 2a+c \ 2c+d = 2b+d \end{array} \]
Simplifying, we get:

• From equation 1: \(2b = 2c \Rightarrow b = c \)
• From equation 2: \(2a = 2d \Rightarrow a = d \)

So, the conditions for which the matrices A and B commute are that b must equal c, and a must equal d. These solutions show how interconnected variables must align to meet specific criteria in systems involving matrices, significantly aiding solving linear algebra problems.