Tiêu đề Solving system of linear equation_01.pdf ✅

Page 2 of 9 Solving a system using Gaussian Elimination & Back Substitution Let us consider the following system of linear equations: We will find the solutions by simplifying the above system through row operations. We can consider the following techniques of row operations: 1. Multiply one equation by a nonzero constant 2. Add one equation to another. We will describe a technique that will help us to perform row operations more systematically and with greater clarity. This particular technique is known as Gaussian Elimination method, which can also be stated as row echelon form. The Augmented Matrix of the above linear system is denoted by: ( | ). In this case “Augmented” indicates to tag something. The Coefficient Matrix of the above system is denoted by: ( ).
Page 3 of 9 Pivot Position: A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the echelon form of A. A pivot column is a column of A that contains a pivot position. If there is a row of all zeros, then it is at the bottom of the matrix. The first non-zero element of any row is a 1. That element is called the leading 1. The leading 1 of any row is to the right of the leading 1 of the previous row. Considering an example: [ ] Pivot Columns The matrix A is in echelon form and thus reveals that columns 1, 2, and 5 are pivot columns. The pivots in this example are 1, 2, and 5. In the Gaussian Elimination Method, the structure of the matrix after elimination should be: [ ] Process: 1. If then we have to interchange with any other row. [ ] [ ] 2. If , then it is either or any other numerical value known as Pivot. Ex: [ ] [ ]
Page 4 of 9 3. If there exists any row that is entirely zero, should be shifted at the bottom. ⌊ ⌋ * + 4. All the elements under each pivot should be . [ ] Let us consider the given exam again. We will solve . ( ) ( ) ( ) We will start with Augmented Matrix of the system: ( | ) ( | ) ( | ) Our pivots are 1, 2 and 5. Now we will solve the system in reverse order since the system is a triangle. We call it Back Substitution.
Page 5 of 9 Here ( ) and ( ). After the row elimination, the matrix becomes and becomes . Let’s write the matrix in terms of system of linear equation: Solution of the system ( ) ( ) Row Echelon Form Of Matrix U ( ) Row Echelon Form: ( ) Elementary row operations & their corresponding Elementary matrices Consider the upper triangular matrix that we have evaluated after row operation: ( ) . It contains three pivots: 1, 2 and 5. The whole purpose of this elimination was to get from to . Elementary Matrix “E”: It is the matrix or product of two or more matrices that takes us from to .