Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a System of Linear Equations (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. $$\left\\{\begin{array}{rr} x_{1}+x_{2}-3 x_{3}= & 9 \\ -x_{1}+2 x_{2} & =6 \\ x_{1}-x_{2}+x_{3} & =-5 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(X = \begin{bmatrix} 4\\ 5\\ -4 \end{bmatrix}\)

Step 1: Setting up the Matrix

Write the system of equations as a matrix (A) and a column vector (B). Here, \(A = \begin{bmatrix} 1 & 1 & -3\\ -1 & 2 & 0\\ 1 & -1 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 9\\ 6\\ -5 \end{bmatrix}\), hence \(A X = B\).

Step 2: Augmenting the Matrix

Set up the augmented matrix [A:B] for performing Gauss-Jordan elimination \(= \begin{bmatrix} 1 & 1 & -3 & 9\\ -1 & 2 & 0 & 6\\ 1 & -1 & 1 & -5 \end{bmatrix}\).

Step 3: Performing Gauss-Jordan Elimination

Start by adding the first row to the second to eliminate the negative term in the second row. This yields \( \begin{bmatrix} 1 & 1 & -3 & 9\\ 0 & 3 & -3 & 15\\ 1 & -1 & 1 & -5 \end{bmatrix}\). Continue by subtracting the first row from the third, giving \( \begin{bmatrix} 1 & 1 & -3 & 9\\ 0 & 3 & -3 & 15\\ 0 & -2 & 4 & -14 \end{bmatrix}\). Next, multiply the second row by 1/3 and add it to the third row, resulting in \( \begin{bmatrix} 1 & 1 & -3 & 9\\ 0 & 1 & -1 & 5\\ 0 & 0 & 2 & -8 \end{bmatrix}\). Finally, multiply the third row by 1/2 and subtract the second row from the first, yielding the row-reduced echelon form \( \begin{bmatrix} 1 & 0 & -2 & 4\\ 0 & 1 & -1 & 5\\ 0 & 0 & 1 & -4 \end{bmatrix}\). This, then, gives the solution matrix \(X = \begin{bmatrix} 4\\ 5\\ -4 \end{bmatrix}\).

Step 4: Checking the solution using graphing utility

A graphing utility can be used to check the validity of the solution. Enter the original equations into the graphing utility and set the three variables, \(x_1\), \(x_2\), and \(x_3\), to their corresponding values in \(X\). If the equations are satisfied, the solution is valid.

Key Concepts

Matrix Equations

A matrix equation is a representation of a system of linear equations using matrix notation. In this formulation, the coefficients of the variables become elements in a matrix, the variables become a column vector, and the constants from the right-hand side of the equations become another column vector. For example, the system of equations presented can be written in matrix form as \( A X = B \), where \( A \) is the coefficient matrix, \( X \) is the column vector of variables, and \( B \) is the column vector of constants.

This compact form is particularly useful because it allows for applying matrix operations to solve the system efficiently. Furthermore, the matrix equation format is closely compatible with computer algorithms, which makes it practical for numerical analysis and solution of even very large systems of equations.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithmic process used to solve systems of linear equations. It involves performing a series of row operations on an augmented matrix to transform it into a row-reduced echelon form. These operations include row swapping, multiplication of a row by a non-zero scalar, and adding a multiple of one row to another row.

The ultimate goal of Gauss-Jordan elimination is to get a diagonal of ones and zeros in all other entries of the leading coefficient's column, thus simplifying the solution process. After obtaining the row-reduced echelon form of the augmented matrix, one can read off the solutions to the variables directly from the matrix. This method is systematic and works for any size of a square system, as long as the system has a unique solution.

Augmented Matrices

An augmented matrix is essentially the coefficient matrix of a system of linear equations combined with the constants column vector. It is written in a block form where the two matrices are adjacent to each other, separated by a line or bracket to denote their boundary. The format is \( [A: B] \), where matrix \( A \) contains the coefficients of the variables and matrix \( B \) contains the constants.

Augmented matrices are pivotal in linear algebra methods, such as Gauss-Jordan elimination, because they hold all of the information necessary to perform row operations without losing the connection between an equation's coefficients and its constant term. The use of augmented matrices streamlines the computational process and makes it less error-prone, which is particularly helpful for complex or large-scale systems.

Row-Reduced Echelon Form

The row-reduced echelon form (RREF) of a matrix is a special form that is achieved through Gauss-Jordan elimination. In RREF, the first non-zero entry in each row, called a leading one, is 1, and it is the only non-zero entry in its column. Furthermore, each leading one is to the right of the leading one in the row above it, and all rows consisting entirely of zeros are at the bottom of the matrix.

Achieving the RREF makes it simple to solve a system of linear equations, as back substitution becomes straightforward or unnecessary. The RREF of an augmented matrix presents the solution directly or indicates if the system is inconsistent or has infinitely many solutions. Understanding RREF is critical for anyone studying linear algebra, as it is a fundamental concept for solving systems of linear equations.