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176 ALGEBRA

For a matrix of size 3×3 (n = 3), the third-order determinant,

a 11 a 12 a 13

Δ ≡ det A ≡ a 21 a 22 a 23
a 31 a 32 a 33

= a 11 a 22 a 33 + a 12 a 23 a 31 + a 21 a 32 a 13 – a 13 a 22 a 31 – a 12 a 21 a 33 – a 23 a 32 a 11 .
This expression is obtained by the triangle rule (Sarrus scheme), illustrated by the following
diagrams, where entries occurring in the same product with a given sign are joined by
segments:

–

+

For a matrix of size n × n (n > 2), the nth-order determinant is defined as follows under
the assumption that the (n – 1)st-order determinant has already been defined for a matrix of
size (n – 1) × (n – 1).
i
Consider a matrix A =[a ij ]of size n × n.The minor M corresponding to an entry a ij
j
is defined as the (n – 1)st-order determinant of the matrix of size (n – 1) × (n – 1) obtained
from the original matrix A by removing the ith row and the jth column (i.e., the row and
i
the column whose intersection contains the entry a ij ). The cofactor A of the entry a ij is
j
i
i
defined by A =(–1) i+j M (i.e., it coincides with the corresponding minor if i + j is even,
j
j
and is the opposite of the minor if i + j is odd).
The nth-order determinant of the matrix A is defined by

a 11 a 12 ··· a 1n
n
n
a 21 a 22 ··· a 2n
Δ ≡ det A ≡ . . . . = a ik A = a kj A .
i
k
. . . k j
. . . . k=1 k=1
.
a n1 a n2 ··· a nn

This formula is also called the ith row expansion of the determinant of A andalsothe jth
column expansion of the determinant of A.
Example 1. Let us find the third-order determinant of the matrix
1 –1 2
( )
A = 6 1 5 .
2 –1 –4
To this end, we use the second-column expansion of the determinant:
3

i+2 i 1+2 6 2+2 1 3+2
det A = (–1) a i2M 2 =(–1) × (–1) × 2 –4 +(–1) ×1 × 2 –4 +(–1) × (–1) × 12
5
2
65
i=1
= 1× [6× (–4)– 5×2]+ 1× [1× (–4)– 2× 2]+ 1× [1×5 – 2× 6]= –49.
5.2.2-2. Properties of determinants.
Basic properties:
1. Invariance with respect to transposition of matrices:
T
det A =det A .
2. Antisymmetry with respect to the permutation of two rows (or columns): if two rows
(columns) of a matrix are interchanged, its determinant preserves its absolute value, but
changes its sign.