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.
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