# Homework #4 (35 pts)

*For ph461 and ph561 (Question 1d is for ph561 students)*

*Due Friday, October 22, 2010 at 5pm*

#### Reading assignment

Chapters 8 and 10.

#### Q1 Properties of Matrices (15 pts)

Consider the matrices

**a)** Verify that Trace(AB) = Trace(BA), det(AB) = det(A)det(B).

**b)** Calculate the commutator [A, B].

**c)** Calculate the inverses A^-1, B^-1, and (AB)^-1. Show that (AB)^-1 = B^-1A^-1.

#### Q1 Additional part required for ph561 students

**d)** Calculate A^2 and infer the expressions of A^(2n) and A^(2n+1). Use these results to calculate the matrix of exp(xA).

#### Q2 Systems of linear equations (10 pts)

Problem 8.36.

#### Q3 Calculating the Laplacian of a scalar field (10 pts)

Problem 10.15.

#### PH505 Students: Reading assignment

Chapter 24.

#### PH505 Students: Additional Homework Question

Problem 24.10.