Row Operations Induction Centre

G'day, and welcome to the Centre!

This centre was established to provide help to visitors who are not familiar with the basic elementary row operations of linear algebra.

Our aim is not to compete with the numerous books that cover this basic material. Rather, our objective is merely to provide free of charge emergency first-aid services for needy persons. If you are really not familiar with the subject we suggest that you consult a book on this subject.

We shall briefly describe the three standard basic elementary row operations as well as the more involved pivot operaton.

These operations are:

- Changy:Interchange of two rows
- Multy: Multiplication of a row by a non-zero scalar
- Addy: Addition of a multiple of a row to another row
- Pivorrratti: The famous povit operation
Contrary to a common misconception, the important thing to observe is that we normally conduct these operations not with a view to torture first year students. Rather, these operations are performed to simplify the content of matrices. Such simplifications are very useful as they enable us to exploit essential properties of certain types of matrices. We shall expand on this point as we introduce the operations.

ChangyThis operation simply involve the interchange of two rows of a matrix. For example, consider the matrix

0 1 0 1 0 0 0 0 1 If we interchange the first two rows, we obtain the following famous matrix:

1 0 0 0 1 0 0 0 1 Why on earth should we be interested in changing the original matrix into this one ? What is wrong with the original matrix and what is so special about the modified matrix?

These are very relevant questions and we suggest that if you are genuinely concerned about these matters that you read an introductory text on linear algebra. A brief discussion on this topic can be found in our discussion of matrix inverse.

As we shall explain below, although on the surface this operation looks more like a military rather than an algebraic operation, the fact is that in the context of numeric matrices it can be formally described by a sequence of algebraic operations.

MultyIt is often necessary and/or desirable and/or convenient to multiply each element of a given row of a matrix by a non-zero scalar. For example, if we multiply the first row of the matrix

3/2 0 0 0 1 0 0 0 1 by 2/3 we obtain the following famous matrix:

1 0 0 0 1 0 0 0 1 Although it is physically possible to multiply a row of a matrix by zero, we normally do not do this. In some cases such an operation is counter-productive, in others it can cause a lot of grief and in still others it may lead to wrong conclusions. For this reason there is an international ban on this operation: the dictum is - Thou shalt not multiply a row of a matrix by zero. So don't ask Multy to do this. He is under strict orders not to perform this operation.

AddyThis is an expansion of Multy: it involves an application of Multy to a given row and is followed by addding the result produced by Multy to another row of the matrix. For example, consider the following matrix:

1 -2 0 0 1 0 0 0 1 If we multiply the second row by 2 and add the result to the first row (element by element), we obtain the folowing famous matrix:

1 0 0 0 1 0 0 0 1 Observe that it does not make sense in this context to multiply a row by zero as the result will be a row of zeros in which case adding it to another row will not change the later. So although it might be fun doing it, from a purely mathematical point of view it is sensless to ask Addy to use a zero as a multiplier. In fact, he will refuse to do it.

PivorrrattiThis operation, commomly called pivoting, is a combination of Multy and Addy: Multy is used on one row and then Addy is used to modify all the other rows of the matrix. The multipliers used by Multy and Addy are chosen so that the resulting operations produce an elementary column - that is a column of zeros, except for one elements which is equal to 1. The location of the "1" in the matrix in the position of the pivot.

Don't panic if you do not see the full picture: the operation is very intuitive and will soon become second nature to you. In fact, experience has shown that students often become addictive to it!. Here is a simple example.

Consider the following matrix:

1 3 0 0 4 0 0 -2 1 Suppose that we wish to transform this matrix into the famous matrix we have met above. One way of doing it is to pivot on the entry "4" as follows: First we apply Multy to the second row using "1/4" as the multiplier. In other words, we divide the second row by 4. This yields the following:

1 3 0 0 1 0 0 -2 1 We now ask Addy to conduct two operations. First multiply the second row by -3 and add the result to the first row. This yields the following:

1 0 0 0 1 0 0 -2 1 Then we ask Addy to multiply the second row by 2 and add it to the third row. This is what we obtain:

1 0 0 0 1 0 0 0 1 you can see why Pivorrratti is so popular: not only that he performs a useful task, you - the user - do'nt have to worry about the multipliers. All you have to tell him is the pivot position - namely the pivot row and pivot column.

Life is beatiful!