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Determinant of a matrix

The determinant of a matrix is one of the main numerical characteristics of a square matrix, used in solving of many problems.
Definition.
Determinant of a matrix A n×n is number:
det(A) = Σ(-1)N(α1,α2,...,αn)·aα11·aα22·...·aαnn
(α1,α2,...,αn)
where (α1,α2,...,αn) - permutation of numbers from 1 to n, N(α1,α2,...,αn) - the number of inversions in the permutation, summation goes over all possible permutations of order n.
Notation
The determinant of a matrix A is usually denoted: det(A), |A| or ∆(A).

Determinant of a matrix - properties

  1. The determinant of a identity matrix is equal to one:

    det(In) = 1

  2. The determinant of a matrix with two equal rows (columns) is equal to zero.
  3. The determinant of a matrix with two proportional rows (columns) is equal to zero.
  4. The determinant of a matrix with a zero row (column) is equal to zero.
  5. The determinant of a matrix is equal to zero if the two or more rows (columns) of this matrix are linearly dependent.
  6. The determinant of a transposed matrix:

    det(A) = det(AT)

  7. The determinant of a inverse matrix:

    det(A-1) = 1det(A)

  8. The determinant of a matrix does not change, if to some of its row (column) to add another row (column) multiplied by some number.
  9. The determinant of a matrix does not change, if to some of its row (column) to add a linear combination of other rows (columns).
  10. If you interchange two rows (columns) of the matrix, the determinant of the matrix changes sign.
  11. The common factor in a row (column) may be taken outside of the determinant:
    a11a12...a1n
    a21a22...a2n
    ....
    k·ai1k·ai2...k·ain
    ....
    an1an2...ann
     = k
    a11a12...a1n
    a21a22...a2n
    ....
    ai1ai2...ain
    ....
    an1an2...ann
  12. det(kA) = kn det(A)       where A is n×n matrix.
  13. If every element in some row of the determinant is equal to the sum of two terms, the original determinant is equal to the sum of two determinants:
    a11a12...a1n
    a21a22...a2n
    ....
    bi1 + ci1bi2 + ci2...bin + cin
    ....
    an1an2...ann
     = 
    a11a12...a1n
    a21a22...a2n
    ....
    bi1bi2...bin
    ....
    an1an2...ann
     + 
    a11a12...a1n
    a21a22...a2n
    ....
    ci1ci2...cin
    ....
    an1an2...ann
  14. The determinant of the upper triangular matrix equal to the product of its diagonal elements.
  15. The determinant of the matrix product equal to the product the determinants of these matrices:

    det(A·B) = det(A)·det(B)


Determinant of a matrix - methods of calculation

Determinant of 1×1 matrix

Rule:
For the matrix of the first order the value of determinant equal to the value matrix element:

∆ = |a11| = a11



Determinant of 2×2 matrix

Rule:
For a matrix of 2×2 the determinant is equal to the difference between the value of products of elements of the main diagonal and antidiagonal:
∆ = 
a11a12
a21a22
 = a11·a22 - a12·a21
Example 1.
Find determinant of a matrix A
A = 
(57)
-41

Solution:

det(A) = 
57
-41
 = 5·1 - 7·(-4) = 5 + 28 = 33



Determinant of 3×3 matrix

Triangle's rule

Rule:
The value of the determinant is equal to the sum of products of main diagonal elements and products of elements lying on the triangles with side which parallel to the main diagonal, from which subtracted the product of the antidiagonal elements and products of elements lying on the triangles with side which parallel to the antidiagonal.
determinant + determinant -
+

∆ = 
a11 a12 a13
a21 a22 a23
a31 a32 a33
 =

a11·a22·a33 + a12·a23·a31 + a13·a21·a32 - a13·a22·a31 - a11·a23·a32 - a12·a21·a33

Sarrus' rule

Rule:
Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from bottom to top:
∆ = 
a11 a12 a13 a11 a12
a21 a22 a23 a21 a22
a31 a32 a33 a31 a32
 =

a11·a22·a33 + a12·a23·a31 + a13·a21·a32 - a13·a22·a31 - a11·a23·a32 - a12·a21·a33

Example 2.
Find determinant of a matrix A
A = 
(571)
-410
203

Solution:

det(A) = 
5 7 1
-4 1 0
2 0 3
 = 5·1·3 + 7·0·2 + 1·(-4)·0 - 1·1·2 - 5·0·0 - 7·(-4)·3 =

= 15 + 0 + 0 - 2 - 0 + 84 = 97




Determinant of n×n matrix

Expanding to Find the Determinant

Rule: Expanding by row
  • Pick any row in the matrix. It does not matter which row you use, the answer will be the same for any row.
  • Multiply every element in that row by its cofactor and add. The result is the determinant:
n
det(A) = Σaij·Aij          - expanding by i row
j = 1
Rule: Expanding by column
  • Pick any column in the matrix. It does not matter which column you use, the answer will be the same for any column.
  • Multiply every element in that column by its cofactor and add. The result is the determinant:
n
det(A) = Σaij·Aij          - expanding by j column
i = 1
Some rows or columns are better than others:
  • Pick the row or column with the most zeros in it. (Since each minor or cofactor is multiplied by the zero element is equal zero)
  • Pick the row or column with the largest numbers (or variables) in it. (The elements in the row or column that you expand along are not used to find the minors. Is easier calculate cofactors with smallest numbers)
Example 3.
Find determinant of a matrix A
A = 
(241)
021
211

Solution: Expand determinant on the first column:

det(A) = 
241
021
211
 =
= 2·(-1)1+1·
21
11
 + 0·(-1)2+1·
41
11
 + 2·(-1)3+1·
41
21
 =

= 2·(2·1 - 1·1) + 2·(4·1 - 2·1) = 2·(2 - 1) + 2·(4 - 2) = 2·1 + 2·2 = 2 + 4 = 6


Example 4.
Find determinant of a matrix A
A = 
(2411)
0200
2113
4023

Solution: Expand determinant on the second row:

det(A) = 
2411
0200
2113
4023
 =
= -0·
411
113
023
 + 2·
211
213
423
 - 0·
241
213
403
 + 0·
241
211
402
 =

= 2·(2·1·3 + 1·3·4 + 1·2·2 - 1·1·4 - 2·3·2 - 1·2·3) = 2·(6 +12 + 4 - 4 - 12 - 6) = 2·0 = 0


Transform matrix to upper triangular form

Rule:
Using the properties of the determinant 8 - 11 for elementary row and column operations transform matrix to upper triangular form. Determinant of of the upper triangular matrix equal to the product of its main diagonal elements.
Example 5.
Find determinant of a matrix A
A = 
(2411)
0210
2113
4023

Solution: transform matrix to upper triangular form

det(A) = 
2411
0210
2113
4023

R3 - R1 → R3 (multiply 1 row by -1 and add it to 3 row); R4 + 2R1 → R4 (multiply 1 row by 2 and add it to 4 row):

det(A) = 
2411
0210
2 - 21 - 41 - 13 - 1
4 - 2·20 - 2·42 - 2·13 - 2·1
 = 
2411
0210
0-302
0-801

C2 ↔ C3 (interchange the 2 and 3 lolumns):

det(A) = -
2141
0120
00-32
00-81

C3 + 8C4 → C3 (multiply 4 column by 8 and add it to 3 column):

det(A) = -
214 + 8·11
012 + 8·00
00-3 + 8·22
00-8 + 8·11
 = -
21121
0120
00132
0001
 = -2·1·13·1 = -26

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