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A system of equations is a collection of two or more equations with the same set of unknowns. The main goal when solving a system of equations is to find the values of the unknown variables that satisfy all equations simultaneously. For example, consider the equations:
- \( 2x - y = 0 \) - \( x + y = 3 \) Here we have two equations with two variables, \( x \) and \( y \). To solve these, we can use various methods, such as substitution, elimination, or matrices. Matrices offer a structured way of finding solutions, especially for larger systems.In matrix form, these equations become:
- Coefficient matrix, \( A = \begin{bmatrix} 2 & -1 \ 1 & 1 \end{bmatrix} \)- Variable matrix, \( \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix} \)- Constant matrix, \( \mathbf{b} = \begin{bmatrix} 0 \ 3 \end{bmatrix} \)This provides a neat alternative to handling systems, capturing all information in a crisp, mathematical framework.

An inverse matrix is like a secret key in matrix algebra. It allows us to reverse the effect of a matrix. If you multiply a matrix by its inverse, you get the identity matrix, much like multiplying a number by its reciprocal gives you one. In terms of solving systems of equations, if we have a system written as \( A\mathbf{x} = \mathbf{b} \), we can find \( \mathbf{x} \) using the inverse matrix \( A^{-1} \). This is done by multiplying both sides by \( A^{-1} \):
\[ \mathbf{x} = A^{-1}\mathbf{b} \]However, not all matrices have an inverse. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible. For our coefficient matrix \( A = \begin{bmatrix} 2 & -1 \ 1 & 1 \end{bmatrix} \), the determinant is non-zero, hence \( A \) has an inverse. Calculating the inverse involves swapping the diagonal elements, changing the signs of the off-diagonal elements, and dividing by the determinant.Knowing how to find an inverse is essential for solving systems efficiently, particularly in higher dimensions.

The determinant of a matrix is a special number that can tell us a lot about the matrix. For square matrices, it is a scalar value that can show us if a matrix is invertible and if a system of equations has a unique solution.For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as:
\[ \det(A) = ad - bc \]In the given exercise, the coefficient matrix \( A = \begin{bmatrix} 2 & -1 \ 1 & 1 \end{bmatrix} \) has a determinant of \( 3 \). This non-zero determinant is what tells us the matrix is invertible, meaning there's a unique solution to the system of equations.The determinant is crucial in determining if there are any solutions:

• If the determinant is zero, there might be no solution or infinitely many solutions.
• A non-zero determinant indicates a single, unique solution.

Knowing and calculating determinants is vital for matrix algebra and solving linear systems.