MTH 2201 Florida Institute of Technology Differential Equations Questions

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Differential Equations/Linear Algebra
Graded Homework – I
MTH 2201/2202
05/17/2021
Note: The answers to the following problems are to be submitted latest by
10.00 a.m. on 06/01/2021. The answers should be written legibly on a
regular size copy paper. You must show all your work to receive full credit.
Late submissions have a 5 point penalty. Any submission made 24
hours beyond the deadline will have a 10 point penalty.
1. Find the value(s) of k so that u = (2, 3k, −4, 1, 5), v = (0, −k, −1, 0, 1)
are orthogonal. Use the value(s) of k you found and normalize the
vectors u, v.
2. Let z1 = 9 + 2i, z2 = 3 − 5i. Find z1 z2 ,
z1
z2
and |z1 z2 |.
3. Determine whether the vector w is a linear combination of the vectors
v1 , v 2 . 



 
−2
1
−4
;
; v2 =
; v1 =
(i) w =
3
1
11




 
3
−2
1
;
; v2 =
; v1 =
(i) w =
−6
4
1

4. Let A =
1 2
3 −1

. Find (i) A2 (ii) A−1 (iii) AAT (iv) trace of AAT
5. Consider the following system of linear equations in unknowns x, y, z:
ax + y + z = b,
x + ay + z = 0,
x + y + az = 1.
(a) For which values of a does the system have a unique solution?
(b) For what values of a and b is the system inconsistent?
(c) For which pair of values (a, b) does the system have more than
one solution? How many solutions in fact?
6. Consider the following system of equations:
2×1 − 4×2 + 3×3 − x4 + 2×5 = 0,
3×1 − 6×2 + 5×3 − 2×4 + 4×5 = 0,
5×1 − 10×2 + 7×3 − 3×4 + 18×5 = 0.
Find the general solution of the system.
7. Using matrices A and B given below, find real numbers a, b and c
which satisfy AT = 5B −1 :




1 0 −2
21
−11
2a − b + c

−7
6
A = −1 1 4  ,
B = 3(a + b + c)
2 6 3
−2
2(b + c) − a
−1
8. Compute the following determinants:
6
2
2
1
1
(c) 1
3
0
−1 −1

1
1
9. Find the eigenvalues and eigenvectors of the matrix 
1
−1
0 5 2
(a) −3 4 0 ,
2 5 1
a
a
(b)
a
b
a
a
b
a
a
b
a
a
b
a
,
a
a
1
0
5
1 −2 1
2 −2 3 .
2
3 −1
−3 4
2

1
1 −1
1 −1 1 
.
−1 1
1
1
1
1
10:10
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