Linear Algebra

58 Chapter 1. Linear Systems
2.8 Example We can decide if matrices are interreducible by seeing if Gauss-
Jordan reduction produces the same reduced echelon form result. Thus, these
are not row equivalent � 1 −3
−2 6
� � 1 −3
−2 5
because their reduced echelon forms are not equal.
�
1
�
−3
�
1
�
0
0 0 0 1
2.9 Example Any nonsingular 3×3 matrix Gauss-Jordan reduces to this.
⎛
1
⎝0 0
1
⎞
0
0⎠
0 0 1
2.10 Example We can describe the classes by listing all possible reduced echelon
form matrices. Any 2×2 matrix lies in one of these: the class of matrices
row equivalent to this,
� �
0 0
0 0
the infinitely many classes of matrices row equivalent to one of this type
� �
1 a
0 0
where a ∈ R (including a = 0), the class of matrices row equivalent to this,
� �
0 1
0 0
and the class of matrices row equivalent to this
� �
1 0
0 1
(this the class of nonsingular 2×2 matrices).
Exercises
� 2.11 Decide if the matrices are row equivalent.
� � � � � � �
1 0 2 1
1 2 0 1
(a) ,
(b) 3 −1 1 , 0
4 8 1 2
5 −1 5 2
� �
2 1 −1 � � �
1 0 2
1 1
(c) 1 1 0 ,
(d)
0 2 10
−1 2
4 3 −1
� � � �
1 1 1 0 1 2
(e)
,
0 0 3 1 −1 1
�
0 2
2 10
0 4
� �
1 0
,
2 2
3
2
�
−1
5
2.12 Describe the matrices in each of the classes represented in Example 2.10.
2.13 Describe all matrices in the row equivalence class of these.
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