91Ó°ÊÓ

Determinants are a special number calculated from a square matrix. For a 2x2 matrix, the determinant is computed as follows: \ For matrix \ \ \[ A = \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \] \ \ The determinant of matrix \( A \) is denoted \( \text{det}(A) \) or \( |A| \) and is given by: \ \ \[ \text{det}(A) = ad - bc \] \ In our exercise, matrix \( A \) was \ \[ \begin{pmatrix} 2 & 9 \ 4 & -3 \end{pmatrix} \] \ By formula, we computed: \ \[ \text{det}(A) = (2 \cdot (-3)) - (9 \cdot 4) = -6 - 36 = -42 \] \ This determinant plays a crucial role in solving systems of linear equations using Cramer’s Rule. Without a non-zero determinant, Cramer's Rule cannot be applied.

A system of linear equations is a collection of one or more linear equations involving the same set of variables. Here, we had the following system: \ \ \[ \begin{aligned} 2x + 9y &= -2 \ 4x - 3y &= 3 \end{aligned} \] \ The goal is to find the values of \( x \) and \( y \) that satisfy both equations simultaneously. This type of problem can be effectively solved using matrix methods like Cramer's Rule, especially because these methods clearly organize the coefficients and constants in a structured format.

Matrix algebra is a branch of algebra that deals with matrices and operations such as addition, subtraction, and multiplication of matrices. In this exercise, matrix algebra helped to reformulate the system of equations in a compact matrix form. \ \ The given system of equations was converted to: \ \ \[ A \mathbf{x} = \mathbf{b} \] \ \ Here, \ \ \[ A = \begin{pmatrix} 2 & 9 \ 4 & -3 \end{pmatrix}, \mathbf{x} = \begin{pmatrix} x \ y \ \end{pmatrix}, \mathbf{b} = \begin{pmatrix} -2 \ 3 \end{pmatrix} \] \ \ Each element in the matrix \( A \) represents a coefficient from the system of linear equations, while \( \mathbf{x} \) and \( \mathbf{b} \) consist of the variables and constants, respectively. This matrix formulation allows for the utilization of powerful matrix operations and rules such as Cramer’s Rule to find the solution efficiently.