Problem 52 - College Algebra and Calculus: An Applied Approach

Question

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 0 & \vdots & 3 \\ 0 & 1 & 0 & \vdots & -1 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The given system of linear equations represented by the augmented matrix has solutions as \(x = 3, y = -1, z = 0\).

Step 1: Identify the reduced matrix as system of equations

The given matrix can be written as a set of linear equations: \[x = 3\] \[y = -1\] \[z = 0\]. Each row represents an equation where the variable coefficients are the elements before the vertical dots (vdots), and the constants are the elements after the vdots.

Step 2: Write down the solutions

From the equations obtained in Step 1, the solutions can be directly seen as: \(x = 3, y = -1, z = 0\) .

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used to represent a system of linear equations compactly. In essence, it takes the coefficients of the variables from the equations and aligns them into a rectangle of numbers, creating the matrix portion, followed by a vertical line (often represented by \texttt{\vdots}, called an augmentation bar). To the right side of this line, the constants from each equation are listed, forming the augmented column. For example, consider a simple system of equations:

\[ \begin{array}{ccc|c} a & b & c & d \ e & f & g & h \ i & j & k & l \end{array} \]
Here, for three linear equations with variables x, y, z, the coefficients a, b, c, and so on, correspond to their respective variables. The column 'd', 'h', and 'l' are the constants or outcomes of each equation after the equals sign. This layout is particularly useful when employing techniques like Gauss-Jordan elimination to find algebraic solutions to the system.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For instance, equations that can be graphed as straight lines in a coordinate system are considered linear. Systems like these can have one solution (a single point of intersection), no solution (parallel lines that never meet), or infinitely many solutions (when the equations describe the same line). In algebra, we often find the solution by looking for the values of variables that satisfy all equations simultaneously. To illustrate, the equations from the augmented matrix in our original exercise represent such a system with unique solutions for each variable: x, y, and z.

Matrix Reduction

Matrix reduction, particularly in the form of row reduction, is a sequence of operations applied to a matrix to simplify its rows into a more solvable form. Typically, this process involves three types of operations: swapping the positions of two rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row to another. When applying Gauss-Jordan elimination, the goal is to transform the matrix into reduced row echelon form (RREF). In RREF, each leading entry (the first nonzero number from the left in a row that hasn't been a leading entry in a higher row) is 1, all leading entries are to the right of those in preceding rows, and all columns containing leading entries have zeros in all other positions. The reduced augmented matrix in the original problem is a perfect example of RREF, which makes it straightforward to extract the solution.

Algebraic Solutions

Algebraic solutions refer to the values of variables that satisfy all equations in a system. Upon reducing an augmented matrix to RREF using Gauss-Jordan elimination, the solutions can often be read off directly. In our example, the reduced matrix is equivalent to the equations x = 3, y = -1, and z = 0. The neat arrangement of variables in the matrix simplifies the process of finding their values without lengthy substitution or elimination processes typically used in algebra. It's a direct path to the solution—each variable is essentially isolated on one side of an equation, mirroring the ways a mathematician wants the variables when solving algebraically.