Lecture Notes in Differential Equations - Bruce E. Shapiro

120 LESSON 14. REVIEW OF LINEAR ALGEBRA
Definition 14.14. The determinant of a square matrix is defined as
follows. Let A be a square matrix and let n be the order of A. Then
1. If n = 1 then A = [a] and det A = a.
2. If n ≥ 2 then
det A =
n∑
a ki (−1) i+k det(A ′ ik) (14.18)
i=1
for any k = 1, .., n, where by A ′ ik
we mean the submatrix of A with the
i th row and k th column deleted. (The choice of which k does not matter
because the result will be the same.)
We denote the determinant by the notation
a 11 a 12 · · ·
detA =
a 21 a 22 · · ·
∣ .
∣
(14.19)
In particular, ∣ ∣∣∣ a b
c d∣ = ad − bc (14.20)
and ∣ ∣∣∣∣∣ A B C
∣ ∣∣∣
D E F
G H I ∣ = A E
H
∣
F
∣∣∣ I ∣ − B D
G
∣
F
∣∣∣ I ∣ + C D
G
E
H∣ (14.21)
Definition 14.15. Let v = (x, y, z) and w = (x ′ , y ′ , z ′ ) be Euclidean 3-
vectors. Their cross product is
i j k
v × w =
x y z
∣x ′ y ′ z∣ = (yz′ − y ′ z)i − (xz ′ − x ′ z)j + (xy ′ − x ′ y)k (14.22)
Theorem 14.16. Let v = (x, y, z) and w = (x ′ , y ′ , z ′ ) be Euclidean 3-
vectors, and let θ be the angle between them. Then
|v × v| = |v||w| sin θ (14.23)
Definition 14.17. A square matrix A is said to be singular if det A = 0,
and non-singular if det A ≠ 0.
Theorem 14.18. The n columns (or rows) of an n × n square matrix A
are linearly independent if and only if det A ≠ 0.