368 Combinatorics of Compositions and Words
Theorem B.18 Let A be a square matrix. Then
1. det(A
T
)=det(A).
2. If A is invertible, that is, det(A) =0,then(A
T
)
−1
=(A
−1
)
T
Matrices arise naturally in connection with systems of equations. The fol-
lowing theorem provides one way to compute the solution of a system of
equations.
Theorem B.19 (Cramer’s Rule) If Ax =
b is a system of n linear equations
in n unknowns such that det(A) =0, then the system has a unique solution
x
i
=
det(A
i
)
det(A)
for i =1, 2,...,n,whereA
i
is the matrix obtained by replacing
the entries in the i-th column of A by the entries in the vector
b on the right-
hand side of the system of equations.
An important characterization of a matrix is its eigenvalues and the asso-
ciated eigenvectors.
Definition B.20 Avalueλ for which the system of equations
Ax = λx ⇐⇒ (λI −A)x =
0(B.3)
has a nontrivial solution x =
0 is called an eigenvalue of A. The nontrivial
solutions of (B.3) are called the eigenvectors of A corresponding to λ.The
eigenvalues of A are the roots of the characteristic polynomial det(λI −A) of
A.
Example B.21 Let A =
"
13
42
#
. To find the eigenvalues of A we solve the
equation
det(λI −A)=
"
λ − 1 −3
−4 λ − 2
#
=0
which by (B.1) is equivalent to
(λ − 1)(λ − 2) − 12
= λ
2
− 3λ −10 = (λ +2)(λ − 5) = 0.
Thus the two eigenvalues are λ
1
= −2 and λ
2
=5. To find the eigenvectors
of A that correspond to λ
1
, we need to find nontrivial solutions for the system
of equations (λ
1
I − A)x =0, which reduces to solving −x
1
− x
2
=0.Thus
any vector of the form [−t, t]
T
for any t ∈ R is an eigenvector for λ
1
= −2.
Similarly, the eigenvectors for λ
2
=5are of the form [3t, 4t]
T
.
There is a strong relation between the eigenvalues of a matrix A and its
determinant and trace.
© 2010 by Taylor and Francis Group, LLC