Overview. Matrix Solutions to Linear Systems. Threevariable systems. Matrices. Solving a threevariable system


 Ethelbert Leonard
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1 Overview Matrix Solutions to Linear Systems Section 8.1 When solving systems of linear equations in two variables, we utilized the following techniques: 1.Substitution 2.Elimination 3.Graphing In this section we will develop techniques that can be used for larger systems. Threevariable systems A system of linear equations in three variables is in the form Solving a System of 3 Equations with 3 Variables Graphically, you are attempting to find where 3 planes intersect. If you find 3 numeric values for (x,y,z), this indicates the 3 planes intersect at that point. The solution to a threevariable system is an ordered triple (x,y,z). Solving a threevariable system We will utilize matrix techniques. One method involves using row operations, and is done by hand. The other method involves using your graphing calculator. Matrices Matrices are a valuable tool when used to solve systems of linear equations. A matrix is a rectangular array of numbers. The rows of a matrix are horizontal. The columns of a matrix are vertical. The matrix shown has 2 rows and 3 columns 1
2 Systems of Equations as Matrices We can put linear equations into augmented matrices and perform row operations corresponding to the equation manipulations and combinations we have done in the past. If The augmented matrix is a system of linear equations in three variables, then is the augmented matrix for the system. Example 1 Example 2 When you have a matrix, there are operations you can perform on the rows in that matrix: 1.You can swap rows: 2.You can multiply a row by a constant (and put the result in the place of the original): 2
3 3.You can add two rows together and put the result in the place of one of the rows you added (the other row is unchanged): 4. You can do combinations: The Goal Of Use row operations to convert your augmented matrix to row echelon form: Example 3 Which of the following matrices are in rowechelon form? a) b) c) d) On the next slide, you ll identify this form and see why row echelon form is useful. Getting There Is Half The Fun How to convert an Augmented Matrix into row echelon form: 1.Start working with the first column. If there is a 1 at the top of that column, go to Step 2. If there is not a 1 at the top of that column, swap that row with a row that has a 1 in the first position. 2. The next objective is get a 0 in the remaining positions of the first column. Multiply Row 1 by the opposite of the coefficient of the first entry in the second row, then add to Row 2 and put the result in Row Repeat the process with Row 3, again using Row 1. 3
4 4. Now move on to the second column. You want to get a zero in the last position. If there is a 1 in the second position of the second column, multiply Row 2 by the opposite of the entry in the second position of Row 3, then add to Row 3 and put the result in Row If there is not a 1 in the second position of the second column, you will have to be a bit more creative. Gaussian Elimination with Matrices RowEquivalent Operations 1. Interchange any two rows. 2. Multiply each entry in a row by the same nonzero constant. 3. Add a nonzero multiple of one row to another row. Example 4 Perform the matrix row operation and write the new matrix. We can use matrices, applying Gaussian Elimination, to solve linear systems. Example 5 Solve the following system using Gaussian elimination: GaussJordan Elimination Continues the process until there are 1s on the main diagonal and 0s below and above the main diagonal. Such a matrix is said to be in reduced rowechelon form. Either method can be applied to all of our problems. We will just focus on Gaussian Elimination in this course when work is to be done by hand. 4
5 GaussJordan Elimination Example Example: Use GaussJordan elimination to solve the system of equations from the previous example. We continue to perform rowequivalent operations until we have a matrix in reduced rowechelon form. Next, we multiply the second row by 3 and add it to the first row. Writing the system of equations that corresponds to this matrix, we have We can actually read the solution, (2, 1, 3), directly from the last column of the reduced rowechelon matrix. If this method (GaussJordan) is quicker, why would we use Gaussian elimination? Using a matrix to solve a system by Gaussian elimination provides a standard, programmable approach. When computer programs (which may be contained in calculators) solve systems, this is the method utilized! Example 6 Solve the system of equations using Gaussian elimination. 3y z =  1 x + 5y  z =  43x + 6y + 2z = 11 5
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