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Please select an option. View More Purchase Options. Elementary Linear Algebra 7th Edition. See definition on page 6 of the text. Consider the following system of three linear equations with two variables. One system could be. The graphs are misleading because they appear to be parallel, but they actually intersect at , Add 2 times Row 1 to Row 2.
Add 5 times Row 1 to Row 3. Because the matrix is in reduced row-echelon form, you can convert back to a system of linear equations. The matrix satisfies all three conditions in the definition of row-echelon form. Moreover, because each column that has a leading 1 columns one and four has zeros elsewhere, the matrix is in reduced row-echelon form.
The augmented matrix for this system is. Because the matrix is in row-echelon form, you can convert back to a system of linear equations. Gaussian elimination produces the following. Because the matrix is in row-echelon form, convert back to a system of linear equations.
Because the leading 1 in the first row is not farther to the left than the leading 1 in the second row, the matrix is not in row-echelon form. However, because the third column does not have zeros above the leading 1 in the third row, the matrix is not in reduced row-echelon form. Subtracting the first row from the second row yields a new second row.
Adding 3 times the second row to the third row yields a new third row. Using a computer software program or graphing utility, you obtain. So, there are three free variables. The number of variables is two because it is equal to the number of columns of the augmented matrix minus one. If A is the coefficient matrix of a system of linear equations, then the number of equations is three, because it is equal to the number of rows of the coefficient matrix.
The number of variables is also three, because it is equal to the number of columns of the coefficient matrix. Using Gaussian elimination on A you obtain the following coefficient matrix of an equivalent system. From this row reduced matrix you see that the original system has a unique solution. Because the system composed of Equations 1 and 2 is consistent, but has a free variable, this system must have an infinite number of solutions.
Use Gauss-Jordan elimination as follows. Begin by finding all possible first rows [0 0 0], [0 0 1], [0 1 0], [0 1 a], [1 0 0], [1 0 a], [1 a b], [1 a 0],. For each of these examine the possible remaining rows. Reduced row-echelon form of a given matrix is unique while row-echelon form is not.
See also exercise 64 of this section. See Theorem 1. Multiplying a row by a nonzero constant is one of the elementary row operations. Free eTextbook while your book ships Contract starts on the date of product shipment, not on date of purchase. ISBN: Access to this eTextbook, plus our entire library of Cengage eTextbooks. When your class uses a Cengage online homework platform, you always know how you're doing and what you need to study. You'll get instant access to your assigned eTextbook, plus personalized learning tools like flashcards for studying.
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