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
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








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) =

0
0

0

1
1
F15 Final Exam Problem 4
Linear Algebra, Dave Bayer
Test 1
[Reserved for Score]
9 test1a4p4
8 3 4
[4] Find eAt where A is the matrix
A =
1 −3
−2
0
eAt =





 +


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 =










 +





 +







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 =


0
y(0) =  1 
1










 +





 +







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
F15 Final Exam Problem 8
Linear Algebra, Dave Bayer
Test 1
[Reserved for Score]
0 test1a4p8
0 6 7
[8] Solve for z in the system of differential equations
y 00 =
2y 0 + y + z
z 0 = −2y 0 + 2y + z
where
y(0) = y 0 (0) = 0,
z(0) = 1
z(t) =