Row Reduction and Echelon Forms

Row Reduction and Echelon Forms 1 / 29

Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence and uniqueness of solutions 2 / 29

Row Echelon Form Definition A matrix is said to be in row echelon form when: 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (left-most nonzero entry) of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zero. Example 0 0 0 0 0 0. 0 0 0 0 0 3 / 29

Reduced Row Echelon Form (RREF) Definition A matrix is said to be in reduced row echelon form when: 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (left-most nonzero entry) of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zero. 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. 6 / 29

RREF Examples Example 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 / 29

RREF Examples Example 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 8 / 29

RREF Examples Example 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 9 / 29

Uniqueness of RREF Theorem Each matrix is row-equivalent to one and only one matrix in reduced row echelon form. 10 / 29

Pivots Definitions A pivot position is a position of a leading entry in a row echelon form of the matrix. A pivot is a nonzero number that either is used in a pivot position to create 0 s or is changed into a leading 1, which in turn is used to create 0 s. A pivot column is a column that contains a pivot position. Notes There is no more than one pivot in any row. There is no more than one pivot in any column. 11 / 29

Example Row reduce to echelon form and locate the pivot columns. 0 3 6 4 9 1 2 1 3 1 2 3 0 3 1 1 4 5 9 7 12 / 29

Example Row reduce to echelon form and then to reduced echelon form. 0 3 6 6 4 5 3 7 8 5 8 9 3 9 12 9 6 15 13 / 29

Basic and Free Variables A basic variable is any variable that corresponds to a pivot column in the augmented matrix of a system. A free variable is any non-basic variable. Example 1 6 0 3 0 0 0 0 1 8 0 5 0 0 0 0 1 7 x 1 + 6x 2 + 3x 4 = 0 x 3 8x 4 = 5 x 5 = 7 14 / 29

RREF and Solutions of Linear Systems Using the RREF, we can write out the general solution of a linear system in parametric form. Example 1 6 0 3 0 0 0 0 1 8 0 5 0 0 0 0 1 7 x 1 + 6x 2 + 3x 4 = 0 x 3 8x 4 = 5 x 5 = 7 15 / 29

RREF and Solutions of Linear Systems Using the RREF, we can write out the general solution of a linear system in parametric form. Example 1 6 0 3 0 0 0 0 1 8 0 5 0 0 0 0 1 7 x 1 = 6x 2 3x 4 x 2 free x 3 = 5 + 8x 4 x 4 free x 5 = 7 16 / 29

Warning Use only the reduced row echelon form to write out the general solution of a linear system in parametric form. 17 / 29

Free Variables and Uniqueness of Solutions The following linear system 3x 2 6x 3 + 6x 4 + 4x 5 = 5 3x 1 7x 2 + 8x 3 5x 4 + 8x 5 = 9 3x 1 9x 2 + 12x 3 9x 4 + 6x 5 = 15 has augmented matrix that is row equivalent to the echelon form 3 9 12 9 6 15 0 2 4 4 2 6. 0 0 0 0 1 4 We observe that the system is consistent and has free variables x 3, x 4. This system has infinitely many solutions. 18 / 29

Free Variables and Uniqueness of Solutions The following linear system 3x 1 + 4x 2 = 3 2x 1 + 5x 2 = 5 2x 1 3x 2 = 1 has augmented matrix that is row equivalent to the echelon form 3 4 3 0 1 3. 0 0 0 We observe that the system is consistent and has no free variables. This system has a unique solution. 19 / 29

Free Variables and Uniqueness of Solutions Notes A consistent system with free variables has infinitely many solutions. A consistent system with no free variables has a unique solution. 20 / 29

Existence and Uniqueness of Solutions of Linear Systems Theorem 1. A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. In other words, if and only if an echelon form of the augmented matrix has no row of the form [0... 0 b] where b is nonzero. 2. If a linear system is consistent, then the solution set contains either (i) a unique solution (when there are no free variables) or (ii) infinitely many solutions (when there is at least one free variable) 21 / 29

Example What is the largest possible number of pivots a 4 6 matrix can have? 22 / 29

Example What is the largest possible number of pivots a 6 4 matrix can have? 23 / 29

Example How many solutions does a consistent linear system of 3 equations and 4 unknowns have? 24 / 29

Example Suppose the coefficient matrix corresponding to a linear system is 4 6 and has 3 pivot columns. How many pivot columns does the augmented matrix have if the linear system is inconsistent? 25 / 29

Using Row Reduction to Solve Linear Systems 1. Write the augmented matrix of the system. 2. Row reduce the augmented matrix to obtain a row equivalent echelon form. Decide whether the system is consistent. If not, then stop; otherwise, go to the next step. 3. Continue row reduction to obtain the reduced row echelon form. 4. Write the system of equations corresponding to the matrix obtained in step 3. 5. State the solution by expressing each basic variable in terms of the free variables and declare the free variables. 26 / 29

Example Find the general solution of the linear system. x 1 + 2x 3 = 3 2x 1 + x 2 + 7x 3 = 9 3x 2 + 9x 3 = 9 27 / 29

Example Find the general solution of the linear system. x 1 + 3x 3 = 1 2x 1 + x 2 + 7x 3 = 5 3x 2 + 3x 3 = 9 28 / 29

Acknowledgement Statements of results and some examples follow the notation and wording of Lay s Linear Algebra and Its Applications, 4th edition. 29 / 29