DETERMINANTS
1. A permutation of a set of integers {1, 2, 3,..., n} is an
arrangement of these integers in some order with no omissions and no
repetitions. To find all permutations of a set of numbers, we can
use a permutation tree:
1
2
3
← 3 choices
/\
/\
/\
23
13
12
← 2 choices
||
||
||
32
31
21
← 1 choice
3 · 2 · 1 = 6
123, 132, 213,231,312,321
2. An inversion is said to occur in a permutation whenever a larger
integer precedes a smaller one.
321 – 3 inversions 32 21
31
213 – one inversion 21
132 – one
32
231 – 2 inversions, 21 31
312 – 2 inversions
31
32
123 – no inversion
For example, 123 ← no inversions 1 3 2 ← one inversion.213 - one,
321 - three inversions, 231 - 2 inversions
3. A permutation is said to be even if the total number of inversions
is even. It is odd if the total number of inversions is odd.
Example: Consider the permutation 5 2 3 4 1. 51,54,53,52,21,41,31 - 7 inversions, odd permutation
4. The determinant of an n × n matrix A is the sum of all signed
elementary products of A.
Example:
2x2 matrix
𝒂𝟏𝟏 𝒂𝟏𝟐
|A| = |𝒂
| = +𝒂𝟏𝟏 𝒂𝟐𝟐 - 𝒂𝟏𝟐 𝒂𝟐𝟏
𝟐𝟏 𝒂𝟐𝟐
𝒂𝟏𝟏 𝒂𝟐𝟐 - no inversion - even - +
𝒂𝟏𝟐 𝒂𝟐𝟏 - one inversion - odd - -
𝒂𝟏𝟏
|A| = |𝒂𝟐𝟏
𝒂𝟑𝟏
𝒂𝟏𝟐
𝒂𝟐𝟐
𝒂𝟑𝟐
𝒂𝟏𝟑
𝒂𝟐𝟑 |
𝒂𝟑𝟑
123, 321,312,132,231,213
5. If A is a square matrix, then the minor of entry aij is denoted by
Mij and is defined to be the determinant of the submatrix that remains
after the ith row and j th column are deleted from A.
6. The number (−1)i+jMij is denoted by Cij and is called the cofactor of
entry aij .
7. If A is any n × n matrix and Cij is the cofactor of aij , then the matrix is called
the matrix of cofactors from A. The transpose of this matrix is called the adjoint
of A and is denoted by adj(A).
8. If A is an n × n matrix, then the number obtained by multiplying
the entries in any row or column of A by the corresponding cofactors
and adding the resulting products is called the determinant of A,
and the sums themselves are called cofactor expansions of A. That is,
det(A) = a1jC1j + a2jC2j +···+ anjCnj
[cofactor expansion along the jth column]
and
det(A) = ai1Ci1 + ai2Ci2 +···+ ainCin
[cofactor expansion along the ith row]
Properties of Determinants
1. If A is an n × n matrix, then |AT|=|A|.
2. If a row of A consists entirely of zeros, then |A| = 0.
3. If A contains two rows which are equal, then |A|=0.
4. If A is upper triangular, lower triangular or diagonal, then
|A| = a11a22 ··· ann.