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178 ALGEBRA

The integer r satisfying these two conditions is called the rank of the matrix A and is
denoted by r =rank (A). Any nonzero rth-order minor of the matrix A is called its basic
minor. The rows and the columns whose intersection yields its basic minor are called basic
rows and basic columns of the matrix. The rank of a matrix is equal to the maximal number
of its linearly independent rows (columns). This implies that for any matrix, the number of
its linearly independent rows is equal to the number of its linearly independent columns.
When calculating the rank of a matrix A, one should pass from submatrices of a smaller
size to those of a larger size. If, at some step, one finds a submatrix A k of size k × k
such that it has a nonzero kth-order determinant and the (k + 1)st-order determinants of all
submatrices of size (k +1)×(k +1) containing A k are equal to zero, then it can be concluded
that k is the rank of the matrix A.
Properties of the rank of a matrix:
1. For any matrices A and B of the same size the following inequality holds:
rank (A + B) ≤ rank (A)+rank (B).
2. For a matrix A of size m×n and a matrix B of size n×k,the Sylvester inequality holds:
rank (A)+rank (B)– n ≤ rank (AB) ≤ min{rank (A), rank (B)}.
For a square matrix A of size n × n,the value d = n –rank(A) is called the defect of the
matrix A,and A is called a d-fold degenerate matrix. The rank of a nondegenerate square
matrix A ≡ [a ij ]of size n × n is equal to n.
THEOREM ON BASIC MINOR. Basic rows (resp., basic columns) of a matrix are linearly
independent. Any row (resp., any column) of a matrix is a linear combination of its basic
rows (resp., columns).

5.2.2-4. Expression of the determinant in terms of matrix entries.
1 . Consider a system of mutually distinct β 1 , β 2 , ... , β n , with each β i taking one of the
◦
values 1, 2, ... , n. In this case, the system β 1 , β 2 , ... , β n is called a permutation of the set
1, 2, ... , n. If we interchange two elements in a given permutation β 1 , β 2 , ... , β n , leaving
the remaining n –2 elements intact, we obtain another permutation, and this transformation
of β 1 , β 2 , ... , β n is called transposition. All permutations can be arranged in such an order
that the next is obtained from the previous by a single transposition, and one can start from
an arbitrary permutation.
Example 2. Let us demonstrate this statement in the case of n = 3 (there are n!= 6 permutations).
If we start from the permutation 12 3, then we can order all permutations, for instance, like this (we
underline the numbers to be interchanged):
1 2 3 −→ 2 13 −→ 3 1 2 −→ 1 32 −→ 2 3 1 −→ 321.
Thus, from any given permutation of n symbols, one can pass to any other permutation
by finitely many transpositions.
One says that in a given permutation, the elements β i and β j form an inversion if β i > β j
for i < j. The total number of inversions in a permutation β 1 , β 2 , ... , β n is denoted
by N(β 1 , β 2 , ... , β n ). A permutation is said to be even if it contains an even number of
inversions; otherwise, the permutation is said to be odd.
Example 3. The permutation 451 32 (n = 5) contains N(4 513 2)= 7 inversions and is, therefore, odd.
Any of its transposition (for instance, that resulting in the permutation 4 315 2) yields an even permutation.
The nth-order determinant of a matrix A ≡ [a ij ]ofsize n × n can be defined as follows:

Δ ≡ det A = (–1) N(β 1 ,β 2 ,...,β n) a β 1 1 a β 2 2 ... a β n n ,
β 1 ,β 2 ,...,β n
where the sum is over all possible permutations β 1 , β 2 , ... , β n of the set 1, 2, ... , n.