Page 20 - Advanced Linear Algebra

4 Advanced Linear Algebra

are row equivalent , denoted by ( — ) ,
Theorem 0.2 Matrices (Á )  C Á
if either one can be obtained from the other by a series of elementary row
operations.
)
1 Row equivalence is an equivalence relation. That is,
)
a (— (
)
b ( — )¬ )— (
c ) ( — ) , ) —* ¬( —* .
)
2 A matrix is row equivalent to one and only one matrix that is in
9
(
reduced row echelon form. The matrix 9 is called the reduced row
echelon form of . Furthermore,
(
9~ , Ä, (

where , are the elementary matrices required to reduce to reduced row
(
echelon form.
)
3 ( is invertible if and only if its reduced row echelon form is an identity
matrix. Hence, a matrix is invertible if and only if it is the product of
elementary matrices.
The following definition is probably well known to the reader.
Definition A square matrix is upper triangular if all of its entries below the
main diagonal are . Similarly, a square matrix is lower triangular if all of its

entries above the main diagonal are . A square matrix is diagonal if all of its

entries off the main diagonal are .
Determinants
We assume that the reader is familiar with the following basic properties of
determinants.

-
Theorem 0.3 Let ( C Á ²-³ . Then det ²(³ is an element of . Furthermore,
) ²- , ³
1 For any ) C
det²()³ ~ det²(³ det²)³
)
(
2 ( ) is nonsingular invertible if and only if det ² ( ³ £ .
)
3 The determinant of an upper triangular or lower triangular matrix is the
product of the entries on its main diagonal.
)
4 If a square matrix 4 has the block diagonal form
v ) Ä y
x Æ Æ Å {
4~ x {
Å Æ Æ
w Ä ) z block
then det²4³ ~ det²) . ³