F15 Final Exam Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
9 test1a4p1
8 4 3
Test 1
Name
Uni
[1] Solve the following system of equations.

 
w
1 1 1 0  
1
 1 1 1 0  x  =  1 
 y
1 1 1 0
1
z





w
 x
  =
 y
z


 

w
1
1
0
 x
0
 −1
1
  =   + 
 y
0
 0 −1
z
0
0
0

0  
r
0
 s 
0
t
1
F15 Final Exam Problem 2
Linear Algebra, Dave Bayer
[Reserved for Score]
3 test1a4p2
9 6 6
Test 1
[2] Find the 3 × 3 matrix A such that
 
 
1
0
A 1  =  0 ,
1
0


 
1
0
A  −2  =  0  ,
0
0




0
0
A 1  =  1 
−2
−2
A=

1



0 0
0
0
1
1
1 −3 
A =  1  2 1 −3 =  2
7
7
−2
−4 −2
6








F15 Final Exam Problem 3
Linear Algebra, Dave Bayer
[Reserved for Score]
6 test1a4p3
1 0 1
Test 1
[3] Let f(n) be the determinant of the n × n matrix in the sequence
1
1 1
−1 1


1
1 0
 −1
1 1
0 −1 1

1
1
0
 −1
1
1

 0 −1
1
0
0 −1

0
0

1
1
1
1
0
0
 −1
1
1
0

 0 −1
1
1

 0
0 −1
1
0
0
0 −1

Find f(8) .
f(8) =
f(8) = 34

0
0

0

1
1
F15 Final Exam Problem 4
Linear Algebra, Dave Bayer
[Reserved for Score]
9 test1a4p4
8 3 4
Test 1
[4] Find eAt where A is the matrix
A =
1 −3
−2
0
eAt =
λ = −2, 3
e
At
e−2t
=
5
2 3
2 3





 +


e 3t
+
5
3 −3
−2
2
F15 Final Exam Problem 5
Linear Algebra, Dave Bayer
[Reserved for Score]
1 test1a4p5
9 2 9
Test 1
[5] Find An where A is the matrix


2 1 0
A = 1 2 0
1 2 2
An =
λ = 1, 2, 3










 +





 +







An






1 −1 0
1 1 0
0
0 0
n
1n 
3
1 1 0
−1
1 0  + 2n  0
0 0 +
=
2
2
−2 −1 1
1 −1 0
3 3 0
F15 Final Exam Problem 6
Linear Algebra, Dave Bayer
[Reserved for Score]
6 test1a4p6
6 3 0
Test 1
[6] Solve the differential equation y 0 = Ay where


1 1 1
A =  1 1 1 ,
2 0 1
y =
eAt

0
y(0) =  1 
1










 +





 +












4 2 3
5 −2 −3
−1
1 0
e 
1
t
4 2 3  +  −4
7 −3  +  −1
1 0
=
9
9
3
4 2 3
−4 −2
6
2 −2 0
3t
λ = 3, 0, 0


 




5
−5
1
e3t  
1
t
5 +
4 +  1
y =
9
9
3
5
4
−2
F15 Final Exam Problem 7
Linear Algebra, Dave Bayer
[Reserved for Score]
4 test1a4p7
8 1 5
Test 1
[7] Express the quadratic form
2x2 + 2y2 − 2xz + 2yz + 3z2
as a sum of squares of othogonal linear forms.
2
+
2
+
2







1 −1
1
1 −1 −2
2 0 −1
1 1 0
1
2
1 −1  +  1 1 0  +  −1
1
2
1  =  −1
A =  0 2
3
3
1 −1
1
−2
2
4
−1 1
3
0 0 0

λ = 1, 2, 4
2
1
(x − y + z)2 + (x + y)2 +
(x − y − 2z)2
3
3
F15 Final Exam Problem 8
[Reserved for Score]
Linear Algebra, Dave Bayer
test93a4p8
Te s t 9 3
[8] Solve for z in the system of differential equations
y" = 2y'+y + z
z = —2y -[- 2y T z
where
A
f^n
4'
1 1 1
1 O 0
-zz
\
y(0) = q'(0) = 0, z(0) = 1
h'l
1
z
v^-vSA^t= 2--V0+I =3
rsvvt+st = \-lll= 'H 4+o ^3
A -i=ri-ioi
bz'zoi
rst—ilVU=»
^ v,s;t <1,"^
(^-\f=G
f'l uuil (a
C*)ecR
^At -gltgCA-r>F
vf
,
,3
.
o l -