(PHZ) Physics 3113
Mathematical Physics
Florida Atlantic University
Fall, 2009
Review Problems II
Recommended Reading: Chow, Appendix 2; Boas, Sections 3.13.3.
1. (Chow A2.1, p. 540) Solve the system
2x
y + 2z =
2
x + 10y 3z =
5
x +
y + z = 3
of linear equations using Cramers rule.
2. (Chow A2.2, p. 540) Evaluate the determinants
D(a) :=
12
,
43
518
D(b) := 15 3 6
10 4 2
and
D(c) :=
cos sin
,
sin cos
where in the third case is an arbitrary real number.
3. (Chow A2.3, p. 542) Expand the second-order determinant
D :=
ab
cd
using the Laplace development on each of the four rows and columns in the matrix. Show
explicitly that one nds the same overall value in each case.
4. A2.4, (Chow p. 542) Evaluate the determinant
1 30
D := 2 6 4
1 0 2
using Laplace developments on the rst row and on the rst column. Show that you get the
same result in each case.
5. (Boas 3.3.7) Use the seven rules for determinants enumerated in Chows book to show that
1a
a2
1 a a2
1 a bc
2 = (c a)(b a)(c b) 0 1 b + a = (c a)(b a)(c b).
D := 1 b ac = 1 b b
00
1
1 c c2
1 c ab
Do not simply evaluate the determinants in this problem. The point is to practice using the
rules while establishing each equality above.
6. (Boas 3.3.10) A square matrix A is called skew-symmetric if its components obey Aij = Aji .
Show that the determinant of a skew-symetric matrix of odd order must vanish.