Definition.
Determinant of a matrix A n×
n is number:
det(A) =  Σ  (1)^{N(α1,α2,...,αn)}·a_{α11}·a_{α22}·...·a_{αnn} 
 (α_{1},α_{2},...,α_{n})  
where (
α_{1},
α_{2},...,
α_{n})  permutation of numbers from 1 to
n, N(
α_{1},
α_{2},...,
α_{n})  the number of inversions in the permutation, summation goes over all possible permutations of order
n.
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.



+ 

– 
∆ = 
a_{11} 
a_{12} 
a_{13} 
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_{13}·a_{21}·a_{32} 
a_{13}·a_{22}·a_{31} 
a_{11}·a_{23}·a_{32} 
a_{12}·a_{21}·a_{33}
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:
∆ = 
a_{11} 
a_{12} 
a_{13} 
a_{11} 
a_{12} 
a_{21} 
a_{22} 
a_{23} 
a_{21} 
a_{22} 
a_{31} 
a_{32} 
a_{33} 
a_{31} 
a_{32} 

= 
=
a_{11}·a_{22}·a_{33} +
a_{12}·a_{23}·a_{31} +
a_{13}·a_{21}·a_{32} 
a_{13}·a_{22}·a_{31} 
a_{11}·a_{23}·a_{32} 
a_{12}·a_{21}·a_{33}
Example 2.
Find determinant of a matrix
A
A = 
 5  7  1  
4  1  0 
2  0  3 

Solution:
det(A) = 

=
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) =  Σ  a_{ij}·A_{ij}  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) =  Σ  a_{ij}·A_{ij}  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 = 
 2  4  1  
0  2  1 
2  1  1 

Solution: Expand determinant on the first column:
= 2·(1)^{1+1}· 

+ 0·(1)^{2+1}· 

+ 2·(1)^{3+1}· 

= 
= 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 = 
 2  4  1  1  
0  2  0  0 
2  1  1  3 
4  0  2  3 

Solution: Expand determinant on the second row:
= 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 = 
 2  4  1  1  
0  2  1  0 
2  1  1  3 
4  0  2  3 

Solution: transform matrix to upper triangular form
R_{3}  R_{1} → R_{3} (multiply 1 row by 1 and add it to 3 row);
R_{4} + 2R_{1} → R_{4} (multiply 1 row by 2 and add it to 4 row):
det(A) = 
2  4  1  1 
0  2  1  0 
2  2  1  4  1  1  3  1 
4  2·2  0  2·4  2  2·1  3  2·1 

= 

C_{2} ↔ C_{3} (interchange the 2 and 3 lolumns):
C_{3} + 8C_{4} → C_{3} (multiply 4 column by 8 and add it to 3 column):
det(A) =  
2  1  4 + 8·1  1 
0  1  2 + 8·0  0 
0  0  3 + 8·2  2 
0  0  8 + 8·1  1 

=  

= 2·1·13·1 = 26 