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A system of equations comprises multiple equations that share the same variables. In the context of linear algebra, these equations are linear, meaning each equation is a straight line when plotted on a graph. For example, the given system of equations is:\[ \begin{array}{r} x_{1}+x_{2}=4 \ 2x_{1}-x_{2}=2 \end{array} \] The goal is to find a solution, which is a set of values for \( x_1 \) and \( x_2 \) that satisfy both equations simultaneously. Understanding how these equations relate in a matrix form can simplify solving them.Matrix representation plays a crucial role here. This representation translates the problem into a more manageable mathematical structure, where properties and operations, like Cramer's Rule, can be efficiently applied.

The matrix determinant is a special number calculated from a square matrix. For a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is found using the formula:\[ \text{det}(A) = ad - bc \]In the problem, the determinant of matrix \( A \) helps determine whether there is a unique solution to the system. If the determinant is non-zero, the system has a unique solution. In our case, the determinant \( \text{det}(A) = -3 \) is indeed non-zero.Determinants also play a critical role in Cramer's Rule, where they help calculate solutions for the variables directly. Each determinant of matrices with altered columns helps isolate one variable, offering a pathway to the solution.

Linear algebra is a field of mathematics focusing on vector spaces, matrices, and linear mappings. It is a fundamental tool for understanding systems of equations.

• Vectors and matrices are key components. They allow equations to be represented in a compact form where linear transformations and solutions can be efficiently derived.
• Cramer's Rule, as applied in the exercise, leverages linear algebra to solve systems of equations using matrices.

In this domain, the relationships between different mathematical entities are often studied through linear transformations, providing powerful methods to solve complex problems. Understanding these fundamentals allows for solving not just small-scale problems, but also large systems encountered in various scientific fields.