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Matrices addition is a fundamental concept in matrix algebra and linear algebra. Just as you would add ordinary numbers, matrix addition involves adding the corresponding elements from each matrix together to form a new matrix. However, there is an important precondition for matrices addition: the matrices must be of the same dimensions. In other words, they must have the same number of rows and columns.

For example, if we have two matrices of the same size, say Matrix A and Matrix B, with elements \( a_{ij} \) and \( b_{ij} \) respectively, their sum, Matrix C with elements \( c_{ij} \) is calculated as follows:\[ c_{ij} = a_{ij} + b_{ij} \].

To add matrices, go through each corresponding entry one by one and perform the addition:\[ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \.end{bmatrix} \]
For our given exercise, the matrices \(AB\) and \(AC\) are actually vectors here, which are also matrices with just one column. Their addition can only be done because both vectors are the same size (they both have 2 rows). The result of their addition is also a vector (or a 2-dimensional matrix).

Matrix algebra includes operations such as addition, multiplication, and scalar multiplication of matrices, and it's guided by particular rules. A key operation we see in this exercise is matrix multiplication, which is distinctly different from matrices addition. It involves a row from one matrix and a column from another, where we multiply corresponding elements and then sum the products.
For instance, when multiplying a matrix \( A \) with a matrix \( B \) (noted as \( AB \) ), each element of the product matrix is computed as:\[ (AB)_{ij} = \( \sum_{k=1}^{n} a_{ik} b_{kj} \) \]
This implies that the number of columns in the first matrix, \( A \) must equal the number of rows in the second matrix, \( B \) for the multiplication to be performed. If \( A \) is of size \( m \times n \) and \( B \) is of size \( n \times p \) then the resulting matrix \( AB \) will have a size \( m \times p \).

In the given exercise, we first perform matrix multiplication for both \( AB \) and \( AC \) before proceeding to matrix addition. As you can see throughout the steps, the rule of corresponding dimensions is strictly followed, ensuring that the matrix operations conform to the rules outlined in matrix algebra.

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's a foundational subject that not only students of mathematics should understand, but also one that's heavily applied in fields such as engineering, physics, computer science, and economics.

In linear algebra, concepts like matrix addition, matrix multiplication, determinant computation, and finding inverses of matrices create the backbone for solving more complex problems involving systems of linear equations, transformations, and vector spaces. It also explores the properties of vectors and matrices, and forms the theoretical basis for many methods and algorithms used in computational applications.

The problem presented in the exercise involves multiple key operations from linear algebra. We start by finding the product of matrices (matrix multiplication), which represents a way to transform one vector into another within a given vector space. Then, we perform matrix addition to combine those transformations. Understanding these operations allows you to understand linear transformations, solve systems of equations, and work with vectors and planes in multidimensional space.