91Ó°ÊÓ

A system of linear equations is a collection of two or more linear equations involving the same set of variables. When we have a system like this, our goal is to find the specific values of these variables that satisfy all the given equations at once. In practice, each equation represents a straight line, and the solution to the system is the point(s) where these lines intersect.
For example, consider the given system:

• \(6x_1 + 20x_2 = -8\)
• \(2x_1 + 7x_2 = 2\)

In this case, you have two equations involving two variables, \(x_1\) and \(x_2\). To find the solution, you either use substitution, elimination, or more advanced techniques like matrix operations, which we'll discuss further. The ultimate aim is to find the set of values for \(x_1\) and \(x_2\) that satisfies both equations simultaneously.

The inverse of a matrix is a powerful concept in linear algebra. For a matrix, the inverse is akin to the reciprocal of a number. If you multiply a matrix by its inverse, you get an identity matrix, which acts like multiplying a number by 1. Not all matrices have inverses; only square matrices (same number of rows and columns) can have one, and they must have a non-zero determinant.
To find the inverse of a 2x2 matrix\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \],you can use this formula:\[ A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]The term \(ad-bc\) is the determinant of the matrix. It acts as a scaling factor. If the determinant is zero, the matrix does not have an inverse.

Representing a system of linear equations as a matrix equation is a compact and convenient method. This technique involves expressing the system in the form \(AX = B\), where \(A\) is the matrix of coefficients, \(X\) is a column matrix of variables, and \(B\) is a column matrix of constants.
For instance, given the system:

• \(6x_1 + 20x_2 = -8\)
• \(2x_1 + 7x_2 = 2\),

the matrices are:

• \(A = \begin{bmatrix} 6 & 20 \ 2 & 7 \end{bmatrix}\)
• \(X = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\)
• \(B = \begin{bmatrix} -8 \ 2 \end{bmatrix}\)

By solving \(AX = B\), we leverage the power of matrix inversions to find \(X\) more systematically than through basic algebraic methods.

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, the determinant is calculated as \(ad - bc\) for a matrix:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \].Calculating the determinant is crucial when determining whether a matrix has an inverse. If the determinant is zero, the matrix cannot be inverted. It's essentially a measure of the matrix's "invertibility".In the exercise, the determinant of the matrix \(A\) was found as follows:\[ \det(A) = 6 \times 7 - 20 \times 2 = 42 - 40 = 2 \]A non-zero determinant (in this case, 2) indicates that the matrix A is invertible. This sets the stage to use its inverse to solve the system of equations effectively.