Write each system as a matrix and solve it by Gauss-Jordan...

Write each system as a matrix and solve it by Gauss-Jordan elimination. If a system has infinitely many solutions, show a general solution.
$$\left\{\begin{array}{l}
x+2 y-z=3 \\
y+3 z=1 \\
z=-2
\end{array}\right.$$

Key Concepts

Pivot Variables and Free Variables

In the context of elimination methods, pivot variables are those that correspond to the leading ones in the RREF, while free variables are those that are not associated with a pivot. This distinction determines whether the system has a unique solution, infinitely many solutions, or none.

Reduced Row Echelon Form (RREF)

Reduced row echelon form is a standardized form of a matrix where each leading entry is 1 and is the only nonzero entry in its column. Achieving RREF through Gauss-Jordan elimination simplifies the identification of solutions to the system.

Matrix Representation

Matrix representation involves writing a system of linear equations in a compact form using rows and columns. This method organizes the coefficients of the variables, simplifying the process of applying systematic approaches for solving the equations.

Augmented Matrix

An augmented matrix is formed by appending the constant terms of the system of equations as an extra column to the coefficient matrix. This configuration combines the entire system into a single matrix, which facilitates the use of elimination methods.

General Solution for Linear Systems

When a system has infinitely many solutions, the general solution expresses the dependent variables in terms of one or more free parameters. This formulation encompasses all possible solutions to the system, providing a complete description of the solution set.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithmic approach to solving a system of linear equations. It involves applying a series of row operations to transform the augmented matrix into reduced row echelon form (RREF), where the solution can be directly read off.

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