Matrices and Vector Algebra

1. Given the matrix

A =

0 2i −1

−i 2 0

3 0 0

Find the transpose, the adjoint, and the inverse of A. Verify that AA−1 = A−1A = I

2. A thick lens was shown below. The thickness of the lens is d. The radii of curvature of the

left (right) surface of the lens is R1 (R2). The index of refraction of the lens is n.

(a) Establish the ray matrix A of the thick lens.

(b) If the element A21 = − 1

fwhere f is the focal length, find f .

Answer: A =

1− (n− 1)d

nR1

d

n(1− n)

R2

+(1− n)2d

nR1R2

+(1− n)

R1

(1− n)d

nR2

+ 1

3. Find the coordinate transformation matrix R that describes a rotation by 120◦ about an axis

from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis

toward the origin.

Answer: R =

0 0 1

1 0 0

0 1 0

.

4. Show that the scalar triple product

~a · (~b× ~c) =

∣∣∣∣∣∣∣ax ay az

bx by bz

cx cy cz

∣∣∣∣∣∣∣ .This result is interpreted as the volume of a parallelepiped whose edges are given by ~a,~b and ~c

(see the following figure).

1

5. (a) The angular momentum about the origin of a particle of mass m moving with velocity ~v

on a path that is a perpendicular distance d form the origin is given by mvd. Show that if

~r is the position of the particle, the vector ~J = ~r×m~v represents the angular momentum.

(b) Consider a rigid collection of particles rotating about an axis through the origin, the angular

velocity of the collection being represented by ~ω.

i. Show that the velocity of the ith particle is

~vi = ~ω × ~ri,

and that the total angular momentum ~J is

~J =∑

i

mi[r2i ~ω − (~ri · ~ω)~ri].

ii. Show that the component of ~J along the axis of rotation can be written as Iω, where

I =∑

i

miρ2i is the moment of inertia of the collection about the axis or rotation.

Interpret ρi geometrically.

(c) Prove that the total kinetic energy of the particles is Iω2/2.

2