Solve the following questions:
Q1) find the inverse (if exists) for the following matrix:
2 −1 0
𝐴 = (−1 2 −1)
0 −1 2
1 9
2 3
Q2) If 𝐴 = (
) and 𝐵 = (
), what value(s) of 𝑘 that will
−1 1
−3 𝑘
make 𝐴𝐵 = 𝐵𝐴?
Q3) Determine whether each vector can be written as a linear combination
of the vectors in 𝑆 = {(2, −1,3), (5,0,4)}
a) 𝑢 = (1,1, −1)
𝑏 = (32, −1,27)
Q4) Determine whether the set 𝑆 is linearly independent or linearly
dependent
a) 𝑆 = {(2, 1, −2), (−2, −1, 2), (4, 2, −4)}
b) 𝑆 = {2, 2, 𝑥 + 3, 3𝑥 2 }
Q5) Determine whether the following matrices from 𝑀2,2 form a linearly
independent set
1 −1
4 3
1 −8
𝐴=[
],𝐵 = [
],𝐶 = [
]
4 5
−2 3
22 23
Q6) Determine whether V is a vector space. If it is, verify each vector space
axioms; if not, state all vector space axioms that fail:
A) The set of all pairs of real numbers of the form (𝑥, 0) with the
standard operations.
B) The set of all pairs of real numbers of the form (𝑥, 𝑦), where 𝑥 ≥ 0 ,
with the standard operations.