91Ó°ÊÓ

Determinants in algebra play a crucial role when it comes to solving systems of linear equations. Mathematically, a determinant is a special number calculated from a square matrix. In this context, it helps determine if a system of equations has a unique solution.
In our example, the matrix \( A = \begin{pmatrix} -1 & 3 \ 4 & -5 \end{pmatrix} \) is associated with the system of equations we are solving. To find the determinant of this matrix, apply the formula for a 2x2 matrix:

• \( \det(A) = a\cdot d - b\cdot c \)
• Where \(a, b, c,\) and \(d\) are the elements of the matrix \( A \)

Here, the elements are \(-1, 3, 4,\) and \(-5\), leading to \( \det(A) = (-1)(-5) - (3)(4) = 5 - 12 = -7 \).
A non-zero determinant \( \det(A) \) indicates that the system has a unique solution. If it were zero, it would suggest either no solutions or infinitely many solutions.

A system of linear equations consists of two or more equations with multiple variables. These equations represent straight lines in a graph and finding their solution means finding the point(s) where these lines intersect. In the given example, the system is composed of:

• \(-x + 3y = 17\)
• \(4x - 5y = -33\)

Using Cramer's Rule is one method to find the unique solution for \(x\) and \(y\). This rule requires the determinant of the coefficient matrix (\(A\)) and allows the calculation of each variable separately by forming auxiliary matrices (like \(A_x\) and \(A_y\)).
By calculating the determinants of these auxiliary matrices and substituting them into Cramer's formula, we derive the values of \(x\) and \(y\). The solution \((x = -2, y = 5)\) represents the intersection point of the lines formed by the equations.

Matrix representation simplifies solving systems of linear equations by making them compact and organized, which is beneficial in both manual calculations and computer algorithms. A system of equations like:

• \(-x + 3y = 17\)
• \(4x - 5y = -33\)

can be rewritten in matrix form as \(A\mathbf{x} = \mathbf{b}\), where:

• \(A = \begin{pmatrix} -1 & 3 \ 4 & -5 \end{pmatrix}\)
• \(\mathbf{x} = \begin{pmatrix} x \ y \end{pmatrix}\)
• \(\mathbf{b} = \begin{pmatrix} 17 \ -33 \end{pmatrix}\)

This structure allows us to use operations like matrix multiplication and determinant evaluation. The determinant of \(A\) confirms the existence of a unique solution. Matrices \(A_x\) and \(A_y\), which are used in Cramer's Rule, derive from replacing columns of \(A\) with \(\mathbf{b}\), providing the tailored path to find \(x\) or \(y\) individually.