Linear Algebra, 2020a

54 Chapter One. Linear Systems
One of the classes is the cluster of interrelated matrices from the start of this
section sketched above (it includes all of the nonsingular 2×2 matrices).
The next subsection proves that the reduced echelon form of a matrix is
unique. Rephrased in terms of the row-equivalence relationship, we shall prove
that every matrix is row equivalent to one and only one reduced echelon form
matrix. In terms of the partition what we shall prove is: every equivalence class
contains one and only one reduced echelon form matrix. So each reduced echelon
form matrix serves as a representative of its class.
Exercises
̌ 1.8 Use Gauss-Jordan reduction to solve each system.
(a) x + y = 2 (b) x − z = 4 (c) 3x − 2y = 1
x − y = 0 2x + 2y = 1 6x + y = 1/2
(d) 2x − y =−1
x + 3y − z = 5
y + 2z = 5
1.9 Do Gauss-Jordan reduction.
(a) x + y − z = 3
2x − y − z = 1
3x + y + 2z = 0
(b) x + y + 2z = 0
2x − y + z = 1
4x + y + 5z = 1
̌ 1.10 Find the reduced ⎛echelon form of⎞
each matrix. ⎛
⎞
( ) 1 3 1
1 0 3 1 2
2 1
(a)
(b) ⎝ 2 0 4 ⎠ (c) ⎝1 4 2 1 5⎠
1 3
−1 −3 −3
3 4 8 1 2
⎛
⎞
0 1 3 2
(d) ⎝0 0 5 6⎠
1 5 1 5
1.11 Get ⎛the reduced⎞
echelon form ⎛ of each. ⎞
0 2 1
1 3 1
(a)
⎝
2 −1 1
−2 −1 0
⎠
(b)
⎝
2 6 2
−1 0 0
̌ 1.12 Find each solution set by using Gauss-Jordan reduction and then reading off
the parametrization.
(a) 2x + y − z = 1
4x − y = 3
(b) x − z = 1
y + 2z − w = 3
x + 2y + 3z − w = 7
(c) x − y + z = 0
y + w = 0
3x − 2y + 3z + w = 0
−y − w = 0
(d) a + 2b + 3c + d − e = 1
3a − b + c + d + e = 3
⎠