(a) write the system of equations as a matrix equation A X=B...

(a) write the system of equations as a matrix equation $A X=B$ and (b) use Gauss-Jordan elimination on the augmented matrix $[A: B]$ to solve for the matrix $X .$ Use a graphing utility to check your solution. $$\left\{\begin{aligned} -x_{1}+x_{2} &=4 \\ -2 x_{1}+x_{2} &=0 \end{aligned}\right.$$

(a) write the system of equations as a matrix equation $A X=B$ and (b) use Gauss-Jordan elimination on the augmented matrix $[A: B]$ to solve for the matrix $X .$ Use a graphing utility to check your solution.
$$\left\{\begin{aligned}
-x_{1}+x_{2} &=4 \\
-2 x_{1}+x_{2} &=0
\end{aligned}\right.$$

Key Concepts

Graphing Utility Verification

Using a graphing utility to verify solutions involves plotting the equations on a coordinate system and checking the intersection points, which correspond to the solutions obtained algebraically. This graphical check serves as a practical means to corroborate the correctness of the solution.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithm that uses elementary row operations to reduce an augmented matrix to reduced row echelon form. This method not only checks the consistency of the system but also directly yields the solution for the system of linear equations, often in a straightforward manner.

Elementary Row Operations

Elementary row operations, which include swapping rows, scaling rows by nonzero factors, and adding multiples of one row to another, are the foundational steps used in methods like Gauss-Jordan elimination. These operations simplify a system while preserving its solution set, ensuring that the transformations lead to an equivalent system.

Matrix Representation of Linear Systems

A matrix equation, typically written as AX = B, is a compact way of representing a system of linear equations where A is the coefficient matrix containing the coefficients of the variables, X is the vector of unknowns, and B is the constants vector. This representation simplifies both notation and the computational process for solving the system.

Augmented Matrix

An augmented matrix is formed by appending the constants column from a system of equations to the coefficient matrix. It is a crucial tool in solving linear systems as it allows one to apply a sequence of row operations systematically, leading to a form where the solutions can be readily identified.

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