18.2. ROW OPERATIONS AND LINEAR EQUATIONS 381

Proof: From the definition of how we multiply matrices, (IA)i j = ∑k IikAk j. Now eachIik = 0 except when k = i when it is 1. Hence the sum reduced so Ai j and so the i jth entryof IA is the same as the i jth entry of A and so IA = A because they are the same matrix. Onthe other side it is similar and this is left as an exercise.

The identity matrix has 1 down the main diagonal and 0 everywhere else. This meansit looks like this in the case of the 3×3 identity: 1 0 0

0 1 00 0 1

It is also standard notation to denote the i jth entry of the identity matrix with the symbolδ i j sometimes δ

ij.

When you multiply by the identity, nothing happens, but when you multiply by anelementary matrix you end up doing a row operation. The next definition is what is meantby an elementary matrix.

Definition 18.2.4 The elementary matrices consist of those matrices which resultby applying a row operation to an identity matrix. Those which involve switching rows ofthe identity are called permutation matrices1.

The importance of elementary matrices is that when you multiply on the left by one, itdoes the row operation which was used to produce the elementary matrix.

Now consider what these elementary matrices look like. First consider the one whichinvolves switching row i and row j where i < j. This matrix is of the form

. . .0 1

. . .1 0

. . .

Note how the ith and jth rows are switched in the identity matrix and there are thus all oneson the main diagonal except for those two positions indicated. The two exceptional rowsare shown. The ith row was the jth and the jth row was the ith in the identity matrix. Nowconsider what this does to a column vector.

. . .0 1

. . .1 0

. . .





...xi...

x j...

=



...x j...xi...

1More generally, a permutation matrix is a matrix which comes by permuting the rows of the identity matrix,

which means possibly more than two rows are switched.