• Elementary Row Operations

I am studying for an exam tomorrow and this is one of the problems given. The instructor gave the solution but I do not understand how he found the solution. The question is 'write down the elementary matrices E1, E2 that correspond to the row operations you performed in part (a) in the order you performed them.

Are these matrices 3x3 or 5x5?' In that answer, he says that they are 3x3. The first row of the matrix E1 is 1 0 0. The second row is 1 1 0. The third row is 0 0 1. The first row of the matrix E2 is 1 0 0. The second row is 0 1 0.

The third row is 0 -1 1. How did he get this solution? The first row operation from part a was R1 + R2. The second row operation from part a was -R2 + R3. I do not understand the concept of an elementary matrix.

Matrix Algebra: Table of Contents Introduction.Matrix operations.Echelon matrices.Matrix properties.Matrix inverse.Matrix applications.Appendices.Elementary Matrix OperationsElementary matrix operations play an important role in many matrixalgebra applications, such asand. Elementary OperationsThere are three kinds of elementary matrix operations. Interchange two rows (or columns).

Multiply each element in a row (or column) by a non-zero number. Multiply a row (or column) by a non-zero number and add theresult to another row (or column).When these operations are performed on rows, they are calledelementary row operations; and when they are performed oncolumns, they are called elementary column operations. Elementary Operation NotationIn many references, you will encounter a compact notation to describeelementary operations.

That notation is shown below. Operation descriptionNotationRow operations1. Interchange rows i and jR i R j2. Multiply row i by s, where s ≠ 0sR i - R i3. Add s times row i to row jsR i + R j - R jColumn operations1. Interchange columns i and jC i C j2. Multiply column i by s, where s ≠ 0sC i - C i3.

Add s times column i to column jsC i + C j - C j. Elementary OperatorsEach type of elementary operation may be performed by matrix multiplication,using square matrices calledelementary operators.For example, suppose you want to interchange rows 1 and 2 of MatrixA. To accomplish this, you could premultiplyA by E to produceB, as shown below.R 1 R 2 =EAR 1 R 2 =0 + 20 + 40 + 60 + 10 + 30 + 5R 1 R 2 =246135= BHere, E is an elementary operator. It operates onA to produce the desired interchanged rows inB. Program do wysylania faxu.

What we would like to know, of course,is how to find E. How to Perform Elementary Row OperationsTo perform an elementary row operation on aA, an r x c matrix, take the followingsteps. To find E, theelementary row operator,apply the operation to an r x r.

To carry out the elementary row operation, premultiplyA by E.We illustrate this process below for each of the three types of elementaryrow operations.Interchange two rows. Suppose we want to interchangethe second and third rows of A, a 3 x 2 matrix. Tocreate the elementary row operator E, we interchangethe second and third rows of the identity matrixI 3.100001010I 3EThen, to interchange the second and third rowsof A,we premultiply A by E, asshown below. R 2 R 3 =12345EAR 2 R 3 =1.0 + 0.2 + 0.41.1 + 0.3 + 0.50.0 + 0.2 + 1.40.1 + 0.3 + 1.50.0 + 1.2 + 0.40.1 + 1.3 + 0.5R 2 R 3 =014523.Multiply a row by a number. Suppose we want tomultiply each element in the second row of Matrix Aby 7. Assume A is a 2 x 3 matrix.

Tocreate the elementary row operator E, we multiply eachelement in the second row of the identity matrixI 2 by 7.1001⇒1007I 2EThen, to multiply each element inthe second row of A by 7,we premultiply A by E. 7R 2 - R 2 =EA7R 2 - R 2 =1.0 + 0.31.1 + 0.41.2 + 0.50.0 + 7.30.1 + 7.40.2 + 7.57R 2 - R 2 =012212835.Multiply a row and add it to another row.Assume A is a 2 x 2 matrix. Suppose we want tomultiply each element in the first row of Aby 3; and we want to add that result to the second row ofA. For thisoperation, creating the elementary row operator is a two-step process.First, we multiply eachelement in the first row of the identity matrixI 2 by 3. Next, we add the result ofthat multiplication to the second row of I 2to produce E.1001⇒100 + 3.11 + 3.0⇒1031I 2EThen, to multiply each element inthe first row of A by 3 and add that result to thesecond row,we premultiply A by E.

3R 1 + R 2 - R 2 =10310123EA3R 1 + R 2 - R 2 =1.0 + 0.21.1 + 0.33.0 + 1.23.1 + 1.33R 1 + R 2 - R 2 =0126. How to Perform Elementary Column OperationsTo perform an elementary column operation onA, an r x c matrix, take the followingsteps. To find E, theelementary column operator,apply the operation to an c x c. To carry out the elementary column operation, postmultiplyA by E.Let's work through an elementary column operation to illustrate theprocess. For example, suppose we want to interchangethe first and second columns of A, a 3 x 2 matrix. Tocreate the elementary column operator E, we interchangethe first and second columns of the identity matrixI 2.1001⇒0110I 2EThen, to interchange the first and second columnsof A,we postmultiply A by E, asshown below. C 1 C 2 =AEC 1 C 2 =0.0 + 1.10.1 + 1.02.0 + 3.12.1 + 3.04.0 + 5.14.1 + 5.0C 1 C 2 =103254Note that the process for performing an elementary column operation on anr x c matrix is very similar to the process for performingan elementary row operation.

The main differences are:. To operate on the r x c matrix A,the row operator E is created from an r x ridentity matrix;whereas the column operator E is created from anc x c identity matrix.

Elementary Row Operations

To perform a row operation,A is premultiplied by E; whereasto perform a column operation, A is postmultipliedby E.