## About Elementary Row Operations

There are three fundamentally different elementary row operations we use in the process of row-reducing a matrix to ((fully) reduced) row-echelon form:

(i) to (or from) a given row we may add (or subtract) a real multiple of some

otherrow:

in the resulting matrix, the given row gets replaced by that sum (or difference), while all the remaining rows (including thatotherrow) are just repeated without change;(ii) we may factor out of (each entry in) a given row a fixed non-zero real number:

in the resulting matrix, the given row gets replaced by the result of that division process, while all the remaining rows are just repeated without change;(iii) we may exchange two distinct rows, e.g., putting row 3 in the position of row 17 while putting row 17 in the position of row 3: again, all unaffected rows are just repeated without change.

To indicate replacement of row 5 by the difference (row 5) – 27×(row 3) it is useful to write “Row

_{5}– 27Row_{3}”;

to indicate dividing row 17 by the divisor π it is useful to write “Row_{17}÷ π”; and

to indicate exchanging rows 3 and 17 it is useful to write “Row_{3}↔ Row_{17}”.Q.: Why “useful”? Useful for what? A.: Useful for the calculation of

determinants(Chapter 2). Here’s the story:One practical way of finding the determinant of a square matrix

Ais by row-reducingAuntil either(i) you find a row of nothing but zeroes, or

(ii)Aeither (a) is, or (b) can be reduced to, an Identity matrix.(Recall that an Identity matrix is a square matrix with nothing but ones along the main diagonal, and zeroes everywhere else.)

In case (i) the determinant of the original matrix is zero —

det(A) = 0 ;

in case (ii (a)), whenAisan Identity matrix, the determinant ofAis said to be 1 (det(A) = 1);

in case (ii (b)), i.e., for matricesAthat are not yet, but can be reduced to, an Identity matrix, you must record, foreachrow operation you use in the full reduction, itsmultiplier, by which I mean

and then multiply together all these multipliers — the resulting (non-zero) product is the determinant of

Row OpMultiplierRow _{i}–rRow_{j}+1 Row _{i}÷rrRow _{i}↔ Row_{j}–1 A.

First posted: 4 Feb 2006. Contents © 2006 by Fred E.J. Linton