91Ó°ÊÓ

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In our example, we have two equations with variables \( x \) and \( y \):

• \( 2x - y = 5 \)
• \( x + y = 1 \)

These equations represent lines on a graph. Solving a system of linear equations entails finding the point(s) where these lines intersect. In essence, it's about finding the values of \( x \) and \( y \) that satisfy both equations simultaneously. These equations can be solved using various methods, such as graphing, substitution, elimination, or matrix algebra. Here, we solve them using matrix algebra, which involves representing the system as a matrix equation \( AX = B \). This provides a powerful and systematic way to handle multiple equations and variables at once.

Matrix inversion is a process used to find a matrix that, when multiplied by the original matrix, yields the identity matrix. In the context of solving systems of equations, the inverse matrix is pivotal because it enables us to isolate the variable matrix:\[ A^{-1}A = I \] where \( I \) is the identity matrix, which has 1s on the diagonal and 0s elsewhere. If we find the inverse of the coefficient matrix, denoted as \( A^{-1} \), we can solve the equation \( AX = B \) by applying \( A^{-1} \) to both sides:\[ X = A^{-1}B \]It is essential that the determinant of \( A \) is non-zero for the inverse to exist. For the given system,\( A = \begin{bmatrix} 2 & -1 \ 1 & 1 \end{bmatrix} \), and its inverse is found using the formula for 2x2 matrices. The resulting inverse helps us find the solution for \( X \), which contains the values of \( x \) and \( y \).

Matrix multiplication is a fundamental operation in matrix algebra, enabling us to combine matrices in a meaningful way. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. For our solution, once we find the inverse of the coefficient matrix \( A^{-1} \), we multiply it by the constant matrix \( B \). This involves computing each element of the resulting matrix by taking the dot product of the rows and columns:\[\begin{bmatrix} x \ y \end{bmatrix} = A^{-1} \begin{bmatrix} 5 \ 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} \ -\frac{1}{3} & \frac{2}{3} \end{bmatrix} \begin{bmatrix} 5 \ 1 \end{bmatrix}\]The calculated matrix gives us the values of \( x \) and \( y \). Each element of this product is obtained by summing the products of corresponding entries. Matrix multiplication can seem complex, but with practice, it becomes a straightforward operation that greatly simplifies solving systems of equations.