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MATH 225
HOMEWORK 3
pp. 159 – 161
#11 – 14, 22, 23, 40, 41, 52 (In all cases identify a specific solution and the
general solution of the corresponding homogeneous system.)
pp. 170 – 171
8, 14 – 16, 23 – 25, 29, 31, 35, 36
A. Describe the solutions of the systems represented by the following augmented matrices [
AB
]
and describe the general solution of the corresponding homogeneous system:
i)
1
0
!
1
2
1
0
1
1
1
0
0
0
0
0
0
"
#
$
$
%
'
'
ii)
1
1
0
!
1
0
1
0
0
1
1
0
2
0
0
0
0
1
3
"
#
$
$
%
'
'
iii)
1
1
1
! !
1
1
0
1
1
! !
1
0
0
0
1
! !
1
1
0
0
0
1
!
1
0
"
"
"
"
"
"
"
0
! !
0
1
1
0
0
0
0
!
0
1
1
"
#
$
$
$
$
$
$
%
'
'
'
'
'
'
∈
R
11x12
(
A
is the matrix with all entries on or above the diagonal
equal to1 and all 0's below it, and
B
∈
R
11
alternates its entries between 1 and 0:
A
i
12
= (–1)
i
+1
.
B. Let
A
=
"
2
3
3
0
1
1
0
"
3
"
2
1
2
3
0
1
1
"
2
#
$
%
%
%
'
(
(
(
∈
M
4
(
R
).
What homogeneous system of linear equations describes all
B
∈
R
4
satisfying
AB
=
B
?
Describe all such
B
.
How many free variables are there in the
solution?

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