Problem 1 Which matrices E21, E31 produce zero in the (2, 1) re
spectively (3, 1) position of E21 · A respectively E31 · A for
A = 2
4
2 10
-2 0 1
8 53
3
5
Find the single matrix E that produces both zeros at once and calculate
E · A.
1
Es
.
-
-
[1%7]
and Ese
-
-
[7%7]
Hence
E-
=
! E)
EA
-
-
[1%5]Problem 2 Use the Gauss elimination method in order to fifind the
inverses of the following matrices:
A = 2
6
6
4
0002
0030
0400
5000
3
7
7
5
, B =
2
6
6
4
3200
4300
0065
0076
3
7
7
5
2
÷ : :: ::::oh ÷
.
÷:: ÷: ::l
n
n
E
:
:::!÷÷
"
:
"
!I
.
Hence A:
too
..
÷
"
:
"
i: : ÷ ÷:÷÷÷H÷÷÷÷
.
i
"
÷:
÷
.
is:÷÷÷i÷÷÷÷÷s
.
÷ : ÷:is::÷
.
÷
.
i
n
: ÷ :i
.
÷÷÷÷d
.
.
..
*
.
÷ :Problem 3 [ Factor the symmetric matrix A = 2
4
2 -1 0
-1 2 -1
0 -1 2
3
5 as
A = LDLT .
3
E
: : :D.AE:# It
÷
.
÷÷÷I=E÷÷d
Hence
E¥Eo¥¥.tt:
-
z
O
0
Thus
.
#
t.se
:] % :&
,
.IE!:&]
O
-
43Problem 4 For which vectors b = 2
4
b1
b
2
b
3
3
5
the following system of
equations has a solution A = 2
4
111
001
001
3
5 2
4
x
1
x
2
x
3
3
5
=
2
4
b
1
b
2
b3
3
5 ?
4
A system Ax=b has a
solution if and only if
b belongs to the column space of A
.
Hence
,
be L )
,
S
.Problem 5Reduce the following matrices to their row reduced
echelon form: A = 2
4
12246
12369
00123
3
5, B = 2
4
242
044
088
3
5
5
E. : : : %I
n
E
:: :::
%Eo ::::I
n
E
: : :::I
e: ¥ :s
-
E
o
: :I
n
E
o
:
-
:I
