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5.2. MATRICES AND DETERMINANTS 179

Example 4. Using the last formula, let us calculate the third-order determinant of the matrix from Exam-
ple 1. The numbers β 1, β 2, β 3 represent permutations of the set 1, 2, 3.We have
Δ ≡ det A =(–1) N(1,2,3) a 11a 22a 33 +(–1) N(1,3,2) a 11a 32a 23 +(–1) N(2,1,3) a 21a 12a 33
+(–1) N(2,3,1) a 21a 32a 13 +(–1) N(3,1,2) a 31a 12a 23 +(–1) N(3,2,1) a 31a 22a 13
0 1 1
=(–1) ×1 ×1× (–4)+ (–1) ×1× (–1) ×5 +(–1) ×6 × (–1) × (–4)
2 2 3
+(–1) ×6 × (–1) ×2 +(–1) ×2× (–1) ×5 +(–1) ×2 ×1×2 =–49,
which coincides with the result of Example 1.
2 .The nth-order determinant can also be defined as follows:
◦
n n n

Δ ≡ det A = ··· δ β 1 β 2 ...β n β 1 1 a β 2 2 ... a β n n ,
a
β 1 =1 β 2 =1 β n=1
is the Levi-Civita symbol:
where δ β 1 β 2 ...β n
0, if some of β 1 , β 1 , ... , β n coincide,

= 1, if β 1 , β 1 , ... , β n form an even permutation,
δ β 1 β 2 ...β n
–1, if β 1 , β 1 , ... , β n form an odd permutation.

5.2.2-5. Calculation of determinants.
1 . Determinants of specific matrices are often calculated with the help of the formulas
◦
for row expansion or column expansion (see Paragraph 5.2.2-1). For this purpose, its is
convenient to take rows or columns containing many zero entries.
2 . The determinant of a triangular (upper or lower) and a diagonal matrices is equal to the
◦
product of its entries on the main diagonal. In particular, the determinant of the unit matrix
is equal to 1.
3 . The determinant of a strictly triangular (upper or lower) matrix is equal to zero.
◦
4 . For block matrices, the following formula can be used:
◦
A O A B

B C O C
= =det A det C,
where A, B, C are square matrices of size n × n and O is the zero matrix of size n × n.
5 .The Vandermonde determinant is the determinant of the Vandermonde matrix:
◦
1 1 ··· 1

x 1 x 2 ··· x n
Δ(x 1 , x 2 , ... , x n ) ≡ x 2 1 x 2 2 ··· x 2 = (x i – x j ).

n
. . . . . . .

. . . . 1≤j<i≤n
.
x 1 x 2 ··· x n
n–1 n–1 n–1
5.2.2-6. Determinant of a sum and a product of matrices.
The determinant of the product of two matrices A and B of the same size is equal to the
product of their determinants,
det(AB)= det A det B.

The determinant of the direct sum of a matrix A of size m × m and B of size n × n is
equal to the product of their determinants,
det(A ⊕ B)=det A det B.