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=(
) B=(
) C=(
)
I 4o-i 7 o
Ll Od
±11w 0
ir’--
i (LIL -
1
2.5.7 (Important) If A has row 1 + row 2 = row 3, show that A is not in-
vertible:
(a) Explain why Ax (1, 0. 0) cannot have a solution.
(b) Which right sides (b
,b
1 ,b
2 ) might allow a solution to Ax
3 = b?
(c) What happens to row 3 in elimination?
I 0
A ) -
(i 3 fA) -
T ]i zc}
x f
(io l
f
0A) a (fcc/?
6 qV
(ro /of A )D( C’
(ro 1TA) yDQ
*he 4-t- C0C4/C/
1E Ct
2
2.5.25 Find A-’ and B-’ (if they exist) by elimination of [ A I ] and
[BI]:
/2 1 i\ (2 —1 —1
A=( 121) B=( —1 2 —1
0 0 iJ \\—1 —1 2
/1 I o)
oio)7 -L
1
I— -
oi/ o°I
ILH 1o
( ooLL)
Oo
? (oo
\ 001/ 0 1
E? 4 zoo4J
OO j-1
00
I -i-i fo z —1 —1 I
io/ 2 -IL
\H -1 od/J
z1 -i [0
C 1)
-‘
0 °61H
3
L
3
1211•- B r
J’ 1
0
O0 0
cr
Nc
C o_
cJ) (
E .1,:
I
-d cç
0 zç
0
LI
0
>
0
U
Ct
E 0.0
(ID
7rJ
o o.—
C’!
—
0
C
OQ —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.
(c) If A in invertible then A—’ and A
2 are invertible.
Tr, Ma5 ,7h
df
(11)
()
JVer) A
(A’)(A-y A
5
2.6 Elimination
- = Factorization: A = LU
2.6.3 Forward elimination changes Ax = b to a triangular Ux = C:
y+z=5
2y + 3z = 7
3y + 6z = 11
x+y+ z=5
y +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
in Ax = b by subtracting £3 = times equation 1
and £32 = times the final equation 2. Reverse that
to recover [ 1 3 6 11 ] 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 = LU and b = Lc.
6
2.6.5 What matrix E puts A into triangular form EA = U? Multiply by
1 = L to factor A into LU:
E
/2 1 0
A= ( 0 4 2
6 3 5
0 0
0
/ I °/io
•1 0 1 10 1Q)/o((t)
1 ,1 o /(6J
OLIZ LI)
0ô5
/ 010
100
oI
7
2.6.7 What three elimination matrices ,
21 E
E 32 put A into its upper
31 E
triangular form A 31
2
E 2 = U? Multiply by E’, E’ and Ej’ to
factor A into L times U:
/10 1
A= (2 2 2 L = E’E’E’.
3 4 5
00
( Eo°\r( fI
i o) tt)r/
o 0 )/ I 51
Ii
E -1 ‘°
0
IOO/fo\
•io1 ° I
100 Ii (OoI ( a H / to
OtO
OL
/OtJC/
0 L
OQLJ
O)(
1)
-l L)
C)C)
)/
-
to
() ° °
/ tol
L7
8 15
2.6.13 (Recommended) Compute L and U for the symmetric matrix A:
a a a a
a b b b
A=
a b cc
abcd
Find four conditions on a, b, c, d to get A LU with four pivots.
o cç -
I o o
C
O cb -b
&
U c- c-b
!Oo 1iooo
(# 00 bj -
I f1 0 C 1 1 1 0 Oo c-bc-b
.‘ l
OQO-c)
9
I—
CN ‘
LID
C
CD
— — —
— ‘.—
2
H
CD
CJD
— 0
CD
—
zz
\ ti II
LI
ii
—
jI)
r
H
c:)
°
II
Cl) II
-\
1
— -d
H
- -r U
I —H
S Cy
I
Cl)
o - Ii
Cl) —
ii
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
any permutation. With x = (1. 2,3) and y 4.2) choose P to show
that Px y is not always x Py.
IyI
ie7 ce
I
9
(
T
) r
(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 pu,- ‘
I 4
t*((/ 1?,)
(//1 )- ( ,?
f4
13
2.7.22 Find the PA = LU factorizatiofl (and check them) for
/0 1 1 (/1 2
A= 1 0 1 and A=2 41.
2 3 4 ill
1
o I i o
OoQ1t
7 1 0 / 1
[0 o
O)o)[O\ (ai)
U I I 3 q c) Q 3 Li
° I 10
i o
o ô(oIl
) OH
(
( (00
) = 11
() 7
i oo’IO
/i io)(1 i)
too, LL1)
1zo
oo
0 1
QT QTQ
2.7.40 Suppose equals Q’ (transpose equals inverse, so I):
2
(a) Show that the columns q
,.
1 . , q, are unit vectors: 1.
(b) Show that every two columns of Q are perpendicular: 2
qq 0.
(c) Find a 2 by 2 example with first entry q
11 cos 8.
.e i’ 1y ( T) (1gw I
) -(/4fd)
( (cI(i)Jfl4
I o+
7
‘ A
-
, I
- -
(j
/ )lf)l /
( C05
15
