Midterm Exam 1 - Linear Algebra | MATH 240, Exams of Linear Algebra

Material Type: Exam; Professor: Johnson; Class: INTRO LINEAR ALGEBRA; Subject: Mathematics; University: University of Maryland; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 05/10/2008

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MATH
24O
R
.
Johnson
[151
1
.
Let
E
=
fO
1
-2'
1 1
if
10
1
-3
EXAM
F
=
(
1
0
-3)
Oct.
if,
Perform
the
following
operations
when
possible.
is
not
possible,
explain
why.
If
an
operation
(a)
EF
(
b
)
FE
2
(c)
(d)
F
[2O]
Solve
simultaneously
the
systems
(a)
x
-
2
x
12
x
=
-
2
3
2x
-5x
->-
x
=
1
123
3x
-
7x
1
2
(b)
x
-
2
x
+
x
123
2x
5x
+
x
1
Z
3
3x
-
7x
+
2x
123
E2O]
3
Calculate
the
cofactor
matrix
of
A
='
37
Use
this
to
compute
adj<A)
and
A
-i
[151
if
Wha
ondition
on
a
,
b
,
c
will
ei
e
that
this
system
is
consistent?
x
-»-
2y
-
2z
=
a
y
+
5z
=
b
3x
+
7y
-
z
=
c
[15]
5.
Solve
by
Cramer's
rule
x
+
2x
=
it
2
3
2x
+
ifx
+
3x
=
-2
1
Z
3
3x
+
7x
+
6x
123
0
CONTINUED
ON
OTHER
SIDE
pf2

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MATH 24O R. Johnson

[151 1. Let E =

fO 1 -2' 1 1 if 10 1 -

EXAM

F = ( 1 0 -3)

Oct. if,

Perform the following operations when possible. is not possible, explain why.

If an operation

(a) EF ( b ) FE 2 (c) (d)

F

[2O] Solve simultaneously the systems (a) x - 2 x 12

x = - 2 3 2x -5x ->- x = 1 123 3x - 7x 1 2

(b) x - 2 x + x 123 2x — 5x + x 1 Z 3 3x - 7x + 2x 123

E2O] (^3) Calculate the cofactor matrix of

A ='

37

Use this to compute adj<A) and A

-i

[151 if (^) Wha ondition on a , b , c will ei (^) e that this system is consistent?

x -»- 2y - 2z = a y + 5z = b 3x + 7y - z = c

[15] 5. Solve by Cramer's rule

x + 2x = it 2 3 2x + ifx + 3x = - 1 Z 3 3x + 7x + 6x 123 0

CONTINUED ON OTHER SIDE

f. 15] 6. (a) Suppose A and B are invertible matrices. What is the formula for (AB)" 1?

(b) Give an example of two 2x2 e1ementary matrices A and B for which AB ^ BA.

(c) Suppose A is a square matrix which satisifes A + EA 31.

Show that A is invertible and A = ~ P~ + l~ I. J O

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