MATH1550 Revision exercise 1
1. Let
2 0
A=
,
−1 3
−1 1 −3
B=
.
4 0 2
(a) Find AB and B T A.
A 0 B AT
where I3 is the identity matrix of order 3.
(b) Find
B T I3 I3 0
2. Let


0 1 0
A = 0 0 1 .
0 0 0
(a) Find A2 and A3 .
(b) Find (I3 + A)3 .
(c) Find (I3 + A)100 .
3. Suppose A and B are n × n symmetric matrices. Prove that AB − BA is skewsymmetric.
4. Solve the following systems of linear equations.

 3x1 − 6x2 + 2x3 = 9
−2x1 + 4x2 + x3 = 1
(a)

x1
− 2x2 − 3x3 = −8

3x1 + 6x2 + x3 + 4x4 = 5



−2x1 − 4x2 + 5x3 + 3x4 = 8
(b)
4x1 + 8x2 − x3 + 3x4 = 2


 3x
+ 6x − 2x + x = −1
1
2
3
4
5. Given that the systems of linear equations

 4x1 − 3x2 − x3 = 4
x1
+ 2x2 + 8x3 = −2
 −2x + 3x + ax = −7
1
2
3
has no solution. Find the value of a.
6. Given that the systems of linear equations

2x + 3x2 +


 1
4x1 − x2 −
3x

1 − 2x2 +

 x + 2x +
1
2
has infinitely many solutions.
(a) Find the value of a and b.
(b) Solve the system.
1
7x3
7x3
ax3
5x3
= 11
= 15
= 10
= b
7. Determine whether the following subsets of R3 are vector subspaces.
(a) {[x1 , x2 , x3 ]T |x1 = 0 or x2 = 0}
(b) {[x1 , x2 , x3 ]T |x1 = 0 and x2 + x3 = 0}
(c) {[x1 , x2 , x3 ]T |x21 + x22 = 0}
8. Let A be an m × n matrix. Prove that the set {v ∈ Rm |v = Au for some u ∈ Rn }
form a vector subspace of Rm .
9. Determine whether the following sets of vectors are linearly independent. If the
vectors are linearly dependent, express the zero vector as a linear combination of
the vectors in a non-trivial way.
(a) v1 = [1, 1, 1]T , v2 = [1, 2, 4]T , v3 = [1, 3, 9]T
(b) v1 = [1, 2, 3]T , v2 = [4, 5, 6]T , v3 = [7, 8, 9]T
10. Given v1 = [2, −1, 1]T , v2 = [3, 1, −2]T ,v3 = [4, a, 9]T , v4 = [b, −2, 1]T . It is known
that v1 , v2 , v3 are linearly dependent and v4 lies in the span of v1 , v2 .
(a) Find the value of a and b.
(b) Express the zero vector as a linear combination of v1 , v2 , v3 in a non-trivial
way.
(c) Express 3v3 − v4 as a linear combination of v1 , v2 .
11. Let v1 , v2 , . . . , vk be vectors in Rn and A be an n × n matrix. Prove that if
Av1 , Av2 , . . . , Avk are linearly independent, then v1 , v2 , . . . , vk are linearly independent.
12. Let v1 , v2 , v3 are vectors in Rn . Prove that if v lies in the span of v1 , v2 , v3 , then
v lies in the span of v1 , v1 + v2 , v1 + v2 + v3 .
13. Suppose v1 , v2 , v3 are vectors in Rn such that v1 + v2 , v2 + v3 , v3 + v1 are linearly
independent. Prove that v1 , v2 , v3 are linearly independent.
14. Let a be a real number and v1 = [a, 1, , 1]T , v2 = [1, a, 1]T , v3 = [1, 1, a]T ∈ R3 .
Prove that v1 , v2 , v3 constitute a basis for R3 if and only if a 6= 1, −2.
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