Math 2270 Assignment 4
-
Dylan Zwick
Fall 2012
Section 2.5 1, 7, 25, 27, 29
Section 2.6 3, 5, 7, 13, 16
Section 2.7 1, 12, 19, 22, 40
-
-
-
2.5
-
Inverse Matrices
2.5.1 Find the inverses (directly or from the 2 by 2 formula) of A, B, C:
A=(
)
I 4o-i
Ll Od
ir’-
B=(
7
)
o
0
±11w
i (LIL
-
1
C=(
)
2.5.7 (Important) If A has row 1
vertible:
(a) Explain why Ax
+
row 2
=
row 3, show that A is not in-
(1, 0. 0) cannot have a solution.
) might allow a solution to Ax
3
,b
2
,b
1
(b) Which right sides (b
=
b?
(c) What happens to row 3 in elimination?
A
I
)
0
A
3 fA)
(i
T
-
-
]i
zc}
x
(io
A)
l
f
0
(ro /of A
)D(
1TA)
yDQ
(ro
(fcc/?
qV
6
*he
1E
a
f
Ct
2
C’
4-t-
C0C4/C/
2.5.25 Find A-’ and B-’ (if they exist) by elimination of
[BI]:
(2
B=( —1
\\—1
/2 1 i\
A=( 121)
0 0 iJ
oio)7
oi/
ILH 1o
(
ooLL)
Oo
\
—1
2
—1
/1 I
-L
1
I—
[
A I
]
and
—1
—1
2
o)
-
o°I
? (oo
001/
1
0
E? 4 zoo4J
OO j-1
00
I
-i-i
C
z1
1)
z —1 —1
2
od/J
\H -1
-‘
fo
io/
3
L
-i
-IL
I
[0
0 °61H
1211•3
B r
J’
1
cr
C
cJ)
E
-d
0
LI
>
0
U
Ct
E
(ID
C’!
C
.1,:
O0
o_
0.0
0
(
0
Nc
o o.—
0
cç
0
—
OQ
I
zç
7rJ
0
—oD
2.5.29 True or false (with a counterexample if false and a reason if true):
(a) A 4 by 4 matrix with a row of zeros is not invertible.
(b) Every matrix with l’s down the main diagonal is invertible.
2 are invertible.
(c) If A in invertible then A—’ and A
Tr,
,7h
Ma5
df
(11)
()
JVer)
(A’)(A-y
5
A
A
2.6 Elimination
-
=
Factorization: A
2.6.3 Forward elimination changes Ax
=
=
LU
b to a triangular Ux
=
C:
y+z=5
2y + 3z = 7
3y + 6z = 11
x+y+
y
+
z=5
+2z=2
5z = 6
x+y+z=5
y
+
2z
=
2
z=2
The equation z = 2 in Ux = c comes from the original x+3y+6z = 11
times equation 1
in Ax = b by subtracting £3 =
times the final equation 2. Reverse that
and £32 =
11
recover
1
3
to
6
] in the last row of A and b from the fi
[
nal [ 1 1 1 5 ] and [ 0 1 2 2 ] and [ 0 0 1 2 ] in
U and c:
Row 3 of
[
A b
]
=
( Row 1 + £32 Row 2 + 1 Row 3) of
[Uc].
In matrix notation this is multiplication by L. So A
6
=
LU and b
=
Lc.
2.6.5 What matrix E puts A into triangular form EA
1 = L to factor A into LU:
E
=
U? Multiply by
/2 1 0
A=
0 4 2
6 3 5
(
/ I
10
0
0
0
•1 0
1
1
°/io
1Q)/o((t)
,1 o /(6J
OLIZ
0ô5
/
100
010
oI
7
LI)
21 E
E
31 E
32 put A into its upper
2.6.7 What three elimination matrices ,
2 = U? Multiply by E’, E’ and Ej’ to
31
2
E
triangular form A
factor A into L times U:
A=
/10 1
2 2 2
3 4 5
(
L
(
00
E -1
E’E’E’.
Eo°\r( fI
i o) tt)r/
o 0 )/
I
Ii
‘°
=
IOO/fo\
0
•io1
°
Ii
100
I
(OoI ( a H
0
/OtJC/
OtO
OL
51
L
/ to
O)(
OQLJ
1)
-l
-
L)
C)C)
()
8
to
°
/
tol
L7
15
°
)/
2.6.13 (Recommended) Compute L and U for the symmetric matrix A:
A=
a a a a
a b b b
a b cc
abcd
Find four conditions on a, b, c, d to get A
o
cç
LU with four pivots.
o o
I
-
O cb -b
C
&
U
c-
c-b
1iooo
!Oo
bj
(#
I
00
f1 0
.‘
-
C
l
1 1 1 0
Oo
c-bc-b
OQO-c)
9
zz
—
)
\
ti
II
I—
CN
‘
‘.—
—
ii
—
—
—
—
—
LI
0
CD
0
CJD
CD
2
H
C
CD
LID
C
Cl)
1
—
-
o
Cl)
Cl)
-d
-r
-
c:)
II
-\
ii
S
H
Ii
U
Cy
—
H
I
°
II
r
jI)
I —H
2.7.12 Explain why the dot product of x and y equals the dot product of
Px and Py. Then from (Px)T(Py) = xTy deduce that pTp = I for
4.2) choose P to show
any permutation. With x = (1. 2,3) and y
that Px y is not always x Py.
IyI
ie7
ce
I
r
9
(
T
(P)
)
T
2
T
p
Jo)
15
f
/
oJ
(/ ?
L(
)/)
(z 5 1) f (z tJ
12
ILL/
2.7.19 Suppose R is rectangular (rn by n) and A is symmetric (rn by in).
(a) Transpose RTAR to show its symmetry. What shape is this ma
trix?
(b) Show why RTR has no negative numbers on its diagonal.
T T
AYr)
A
A
)()
(‘)
(ej
afl
3
fM
I
4
t*((/
(//1
)- (
f4
13
pu,-
‘
1?,)
,?
2.7.22 Find the PA
=
LU factorizatiofl (and check them) for
(/1 2
/0 1 1
1 0 1
A=
2 3 4
1
o
7 1 0
[0
O)o)[O\
U I I
3 q
c)
o
(
(
(00
oo
0
1
I 10
ô(oIl
)
=
41.
ill
OoQ1t
°
A=2
and
o
/ 1
o
(ai)
Q 3 Li
)
11
I i
()
i
o
OH
7
i
oo’IO
/i io)(1 i)
too,
LL1)
1zo
2.7.40 Suppose
QT
equals
Q’
(transpose equals inverse, so
,.
1
(a) Show that the columns q
.
,
Q are perpendicular:
11
(c) Find a 2 by 2 example with first entry q
1y ( T)
(
I
7
(1gw
o+
A
-
I
,
-
(j
/
)lf)l
/
(
C05
15
1.
2
qq
0.
cos 8.
(cI(i)Jfl4
‘
I):
2
q, are unit vectors:
(b) Show that every two columns of
.e i’
QTQ
-
I
) -(/4fd)