Raising the bar

Elementary transformations

Here we look at matrix representations of transformations called elementary.

Exercise 1. Let denote the -th unit row vector and let be an arbitrary matrix. Then

a) premultiplication of by cuts out of the -th row

b) Premultiplication of by puts the row into the -th row of the null matrix.

c) Premultiplication of by adds row multiplied by to row without changing the other rows of

d) The matrix has determinant 1.

Proof. a) It's easy to see that

b) Obviously,

( in the -th row, denotes null matrices of conformable dimensions)

c)

d) The matrix has ones on the main diagonal and only one nonzero element outside it. By the Leibniz formula it's determinant is 1.

The reader can easily solve the next

Exercise 2. a) Postmultiplication of by cuts out of the -th column

b) Postmultiplication of by puts the column into the -th column of the null matrix.

c) Postmultiplication of by adds column multiplied by to column without changing the other columns of

d) The matrix has determinant 1.

Exercise 3. a) Premultiplication of by

(1)

permutes rows

b) Postmultiplication of by the transpose of (1) permutes columns

This is a general property of permutation matrices. Recall also that their determinants can be only

Definition. 1) Adding some row multiplied by a constant to another row or 2) adding some column multiplied by a constant to another column or 3) permuting rows or columns is called an elementary operation. Accordingly, matrices that realize them are called elementary matrices.