Row Operations and Augmented Matrices
Learning Outcomes
- Write the augmented matrix for a system of equations.
- Perform row operations on an augmented matrix.
[latex]\left[\begin{array}{ccc|c}\hfill 3& \hfill -1& \hfill -1& \hfill 0\\ \hfill 1& \hfill 1& \hfill 0& \hfill 5\\ \hfill 2& \hfill 0& \hfill -3& \hfill 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 the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.How To: Given a system of equations, write an augmented matrix
- Write the coefficients of the [latex]x-[/latex]terms as the numbers down the first column.
- Write the coefficients of the [latex]y-[/latex]terms as the numbers down the second column.
- If there are [latex]z-[/latex]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: Writing the Augmented Matrix for a System of Equations
Write the augmented matrix for the given system of equations.Answer: The augmented matrix displays the coefficients of the variables and an additional column for the constants.
Try It
Write the augmented matrix of the given system of equations.Answer: [latex]\left[\begin{array}{cc|c}\hfill 4& \hfill -3& \hfill 11\\ \hfill 3& \hfill 2& \hfill 4\\ \end{array}\right][/latex]
[ohm_question]125538[/ohm_question]Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.Example: Writing a System of Equations from an Augmented Matrix Form
Find the system of equations from the augmented matrix.Answer: When the columns represent the variables [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex],
Try It
Write the system of equations from the augmented matrix.Answer: [latex]\begin{array}{c}x-y+z=5\\ 2x-y+3z=1\\ y+z=-9\end{array}[/latex]
Row Operations
Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows. Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner and zeros in every position below the main diagonal as shown.- In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
- Any all-zero rows are placed at the bottom of the matrix.
- Any leading 1 is below and to the right of a previous leading 1.
- Any column containing a leading 1 has zeros in all other positions in the column.
- Interchange rows. (Notation: [latex]{R}_{i}\leftrightarrow {R}_{j}[/latex] )
- Multiply a row by a constant. (Notation: [latex]c{R}_{i}[/latex] )
- Add the product of a row multiplied by a constant to another row. (Notation: [latex]{R}_{i}+c{R}_{j}[/latex])
tip for success
As with other challenging concepts you've encountered in this course, time and patient practice with the process of Gaussian elimination will aid familiarity. If the instructions seem intimidating, work out the example below on paper once or twice, then try to apply the General Note and How To information to the example. Work back and forth between them until you gain a firmer understanding. Then try the practice problems.A General Note: Gaussian Elimination
The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix [latex]A[/latex] with the number 1 as the entry down the main diagonal and have all zeros below.How To: Given an augmented matrix, perform row operations to achieve row-echelon form
- The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary.
- Use row operations to obtain zeros down the first column below the first entry of 1.
- Use row operations to obtain a 1 in row 2, column 2.
- Use row operations to obtain zeros down column 2, below the entry of 1.
- Use row operations to obtain a 1 in row 3, column 3.
- Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only zeros below.
- If any rows contain all zeros, place them at the bottom.
Example: Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form
Perform row operations on the given matrix to obtain row-echelon form.Answer: The first row already has a 1 in row 1, column 1. The next step is to multiply row 1 by [latex]-2[/latex] and add it to row 2. Then replace row 2 with the result.
Try It
Write the system of equations in row-echelon form. [latex-display]\begin{array}{l}\text{ }x - 2y+3z=9\hfill \\ \text{ }-x+3y=-4\hfill \\ 2x - 5y+5z=17\hfill \end{array}[/latex-display]
Answer: [latex-display]\left[\begin{array}{ccc|c}\hfill 1& \hfill -\frac{5}{2}& \hfill \frac{5}{2}& \hfill \frac{17}{2}\\ \hfill 0& \hfill 1& \hfill 5& \hfill 9\\ \hfill 0& \hfill 0& \hfill 1& \hfill 2\\ \end{array}\right][/latex-display]
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