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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}.

1

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