Phys 2920, Spring 2009
Problem Set #3
1. Generally matrices do not commute (for multiplication). The extent to which they do
not is given by the commutator, which for matrices Aand Bis given by
[A,B]≡AB −BA
The following matrices are very useful in physics:
σx= 0 1
1 0 !σy= 0−i
i0!σz= 1 0
0−1!
a) Evaluate each of the three following commutators, and for each express the result in terms
of the σmatrices themselves.
[σx, σy] [σy, σz] [σz, σx]
b) Are the σmatrices symmetric? Orthogonal? Hermitian? Unitary?
2. Show that the matrix
R= cos θ−sin θ
sin θcos θ!
is an orthogonal matrix.
3. Find the determinant of the matrix
A=
5 4 2 1
2 3 1 −2
−5−7−3 9
1−2−1 4
any way you can!
4. For the following matrices, figure out if each has an inverse (give a good mathematical
reason) and then use a computer to find the inverse.
(a) A=
1 3 −4
1 5 −1
3 13 −6
(b) B=
1 2 −4
−1−1 5
2 7 −3
5. Solve the set of equations
x+ 2y−4z=−4
2x+ 5y−9z=−10
3x−2y+ 3z= 11
1
