Description
1. For each of the following statements, produce a counterexample to show that the statement is
false.
(a) If A and B are square matrices, AB = BA.
(b) If AB =
[
1 1
1 1]
, then A and B are 2 × 2 matrices.
(c) If AB = I then BA = I.
(d) If A2 = 0, then A = 0.
2. Let R =
1 2 3
4 5 6
7 8 9
.
(a) Find all solutions to the matrix equation R
x1
x2
x3
=
2
5
8
.
(b) Prove that the set X = {⃗x ∈ R
3
: R⃗x = ⃗0} is a subspace.
3. Suppose E is a 4 × 3 matrix with columns ⃗c1, ⃗c2, ⃗c3 and rows ⃗r1, ⃗r2, ⃗r3, ⃗r4. Let ⃗v =
2
−1
1
.
(a) Express E⃗v as a linear combination of ⃗c1, ⃗c2, ⃗c3.
(b) Supposing ⃗r1 · ⃗v = 1, ⃗r2 · ⃗v = 6, (⃗r3 + ⃗r4) · ⃗v = 2, and (⃗r3 − ⃗r4) · ⃗v = −2, compute E⃗v.
4. Suppose that ⃗u, ⃗v, and ⃗w are vectors in R
2
that are related by the following diagram.
.
4⃗u
2⃗v
3 ⃗w
Let A = [⃗u|⃗v| ⃗w] be the matrix with columns ⃗u, ⃗v, and ⃗w.
(a) What is the rank of A?
(b) Find all solutions to the equation A⃗x = ⃗0.
(c) Find a basis for the subspace V = {⃗x ∈ R
3
: A⃗x = ⃗0}.
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