MATH 223 Assignment #1 (2 pages) Due Friday September 18, 2014 at start of class
x 1
0 1
1. Let A =
and B =
. Determine all x, y so that AB = BA.
1 2
x y
2. Find a 2 × 2 matrix A with A2 = A where A 6= 0, I. Here we use the notation 0 to refer to the
zero matrix, all entries 0. Both 0 and I act as roots to the associated quadratic equation.
3. Gavin has a sum of s dollars in t bills (not treasury bills!) The bills are either $3 or $5. Express
the number x of $3 bills and the number y of $5 bills in terms of s and t.
4. Assume you are given a pair of matrices A, B which satisfy AB = BA. Show that if we set
C = A2 + 2A and D = B 2 − I, then CD = DC. Generalize this, namely find a property so that
for matrices C, D with that certain property, then CD = DC.
5. Let R(θ) denote the matrix of the transformation which rotates the plane by θ counterclockwise
around the origin. Explain in terms of rotations and linear transformations why R(θ)R(φ) =
R(θ + φ). Show how you can use this to derive the high school formulas for cos(θ + φ), sin(θ + φ)
in terms of cos(θ), sin(θ), cos(φ), sin(φ).
6. Let f : R2 → R2 be a linear transformation. We are given that
2
1
3
0
f
=
and f
=
3
0
5
1
Use this to determine f 12 .
7. a) Assume A, B are 2 × 2 invertible matrices so that A−1 and B −1 exist.
Show that (AB)−1 = B −1 A−1 .
b) Given
a b
a c
T
A=
then define A =
,
c d
b d
where AT is called the transpose of A. The dot product of two vectors x = ab , y = dc is
x · y = ac + bd. Using the idea that the i, j entry of AB is the dot product of the ith row of A and
the jth column of B, show that (AB)T = B T AT .
(One could verify (AB)T = B T AT for two arbitrary 2 × 2 matrices A, B directly but your
argument should immediately generalize to larger matrices).
a b
8. Define tr
= a + d (read trace for ‘tr’).
c d
d −b
∗
a) Using A = (a + d)I − A =
we verify AA∗ = (ad − bc)I and the Cayley-Hamilton
−c a
Theorem (for 2 × 2 matrices) that
A2 − tr(A)A + det(A)I = 0,
(i.e. A acts as a ‘root’ in the quadratic det(A − λI) when interpreted as a matrix polynomial).
b) Determine conditions on tr(A) and det(A) to ensure that A2 = A but A 6= I, 0. Hint: Consider
separately the case A 6= kIfor any k and the case A = kI for some k.
c) Given a matrix A = ac db with A2 = A and A 6= I, 0, determine formulas for c, d in terms of a, b.
9. Consider two nonzero vectors x = ab , y = dc . Argue on geometric grounds that there is a θ
with 0 ≤ θ < 2π and a positive constant ρ > 0 so that y = ρR(θ)x. In particular
c
cos(θ) − sin(θ)
a
=ρ
d
sin(θ)
cos(θ)
b
We are interested in whether we are turing left or right when we switch from going in the direction
x to the direction y, i.e. whether θ ∈ (0, π) or θ ∈ (−π, 0) . Now by solving for sin(θ), obtain a
simple inequality (involving a, b, c, d) to determine whether we are going right or left.