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 other row:
in the resulting matrix, the given row gets replaced by that sum (or difference), while all the remaining rows (including that other row) 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 “Row5 – 27Row3”;
to indicate dividing row 17 by the divisor π it is useful to write “Row17 ÷ π”; and
to indicate exchanging rows 3 and 17 it is useful to write “Row3 ↔ Row17”.
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 A is by row-reducing A until either(i) you find a row of nothing but zeroes, or
(ii) A either (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)), when A is an Identity matrix, the determinant of A is said to be 1 (det(A) = 1);
in case (ii (b)), i.e., for matrices A that are not yet, but can be reduced to, an Identity matrix, you must record, for each row operation you use in the full reduction, its multiplier, by which I mean
and then multiply together all these multipliers — the resulting (non-zero) product is the determinant of A .
Row Op Multiplier Rowi – rRowj +1 Rowi ÷ r r Rowi ↔ Rowj –1
First posted: 4 Feb 2006. Contents © 2006 by Fred E.J. Linton