MAT 205 Midterm, Fall 2011
American University of Science and Technology · Linear Algebra · 2 pages · Ms. Hoda Mokbel
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I. (20 Points) Answer by True or False. To obtain complete grade, show all the
calculations and proofs where needed:
1) If the square matrices A and B are symmetric then the matrix (A + B) is
symmetric (……..).
2) Det(4A)= 4Det(A) for all (4 4) matrices (……..).
3) The matrices
8
1
4
3
1
2
2
0
1
A
and
1
1
6
1
0
4
2
2
11
B
are inverses (…….).
4) The determinant of any diagonal (n n) matrix is the product of the entries on the
main diagonal (……..).
5) The matrix
1
1
1
2
1
4
1
2
k
k
is invertible for all positive constants k (……..).
6) The vectors
( , )
v a b
and
, )
(
b a
w
are orthogonal vectors (……..).
7) If
k u
ku
, then
k 0
(……..)
8) The determinant of
A
0
2
0
0
0
0
0
0
6
0
0
0
0
0
4
3
0
0
0
0
0
0
5
0
0
is 0 (……..).
9) The system
3
2
1
0
0
0
6
5
4
3
2
1
z
y
x
is inconsistent (……..)
10) If a matrix A is in reduced row echelon form, then at least one of the entries in
each column must be 1 (…….).
II. (10 Points) Find the Area of the triangle having vertices
;2 3)
A( ;0
,
)1;0
B ;1(
and
0)
;1;2
C(
. Having the area, deduce the length of the altitude from
vertex C to side AB.
III. (15 Points) Consider two distinct numbers, a and b. We define the function:
2
2
2
1
1
1
det
)
(
t
b
a
t
b
a
f t
a) Show that f(t) is a quadratic function (i.e. of degree 2). What is the
coefficient of t2?
b) Explain why f(a) = f(b) = 0. Conclude that
)
)(
(
( )
b
a t
k t
f t
, for
some constant k. Find k using your work in part (a).
c) For which values of t is the matrix invertible?
IV. (20 Points) Let
u (3;2 )1;
,
;3 4)
( ;5
v
and
7)
;6
;1(
w
. Compute the
following:
a)
)
(
w
u v
b)
proj u v
c)
w
v
u
3 5
d)
u w v
.
e)
2 )
(
w
v
u
V. (20 Points) Let the vectors
;1( ;1 )3
u
,
7)
;2
;1(
v
and
w ( ;2 ;3 4)
. Find
scalars
1c ,
2
c , and
3
c such that
( ;8 ;1 10)
3
2
1
c w
c v
c u
VI. (15 Points) Let
3
1
1
1
3
1
1
1
3
A
and
1 1
1
1 1
1
1 1
1
B
a) Show that A can be written as
aI bB
3
, where a, b are scalars to
determine, and I3 is the identity matrix (3 3).
b) Determine
2
B .
c) Determine
2
A and verify that
3
2
10
7
I
A
A
d) Determine
A1