Chapter 2 Matrices
2.2 Solving Systems Using Matrices
Learning Objectives
By the end of this section, you will be able to:
- Write a system of equations as an augmented matrix
- Use a matrix and row reduction (Gaussian elimination) to solve a system of equations
- Interpret the solutions from an augmented matrix
- Recognize dependent and inconsistent systems of equations
While the techniques we learned in the last section can be used to solve any 2-by-2 or 3-by-3 system of linear equations, mathematicians often look for ways to do problems while writing less (which is why we use single letters for variables instead of full words) and to make solving problems more procedural. For systems of linear equations, matrices are the tools we use. In addition to making solving small systems more straightforward, the techniques can be extended to solve 4-by-4, 100-by-100, or even larger systems of linear equations that commonly arise in science.
A matrix is a rectangular array of numbers arranged into rows and columns.
Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
Given a system of equations, write an augmented matrix:
- Write the coefficients of the x-terms as the numbers down the first column.
- Write the coefficients of the y-terms as the numbers down the second column.
- If there are z-terms, write the coefficients as the numbers down the third column.
- Draw a vertical line and write the constants to the right of the line.
Example 1
Consider the following 2×2 system of equations. Write this system as an augmented matrix:
[latex]\begin{cases}3x+4y=7\\4x-2y=5\end {cases}[/latex]
Extract the coefficients from the system and write them in a rectangular array. This is called the coefficient matrix.
[latex]\begin{bmatrix}3&4\\4&-2\end{bmatrix}[/latex]
We then draw a vertical line and extract the constants from the right-hand side of the system equations. This is the augmented matrix for the given system.
[latex]\left[ \begin{array}{cc|c}3&4&7\\4&-2&5\end{array}\right][/latex]
Example 2
Write the augmented matrix for the 3×3 system of equations.
[latex]\begin{cases}3x-y-z=0\\x+y=5\\2x-3z=2\end{cases}[/latex]
The coefficient matrix is
[latex]\begin{bmatrix}3&-1&-1\\1&1&0\\2&0&-3\end{bmatrix}[/latex]
And the system is represented by the augmented matrix
[latex]\left[ \begin{array}{ccc|c}3&-1&-1&0\\1&1&0&5\\2&0&-3&2\end{array}\right][/latex]
Notice that the matrix is written so that the variables line up in their own columns: [latex]x[/latex]-terms go in the first column, [latex]y[/latex]-terms in the second column, and [latex]z[/latex]-terms in