
Determinant of a matrixPage Navigation:
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:
Notation The determinant of a matrix A is usually denoted: det(A), A or ∆(A).Determinant of a matrix  properties
Determinant of a matrix  methods of calculationDeterminant of 1×1 matrixRule: For the matrix of the first order the value of determinant equal to the value matrix element:
∆ = a_{11} = a_{11} Determinant of 2×2 matrixRule: 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:
Example 1. Find determinant of a matrix A
Solution:
Determinant of 3×3 matrixTriangle's ruleRule: 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_{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' ruleRule: 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_{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
Solution:
= 15 + 0 + 0  2  0 + 84 = 97 Determinant of n×n matrixExpanding to Find the DeterminantRule: Expanding by row
Rule: Expanding by column
Some rows or columns are better than others:
Example 3. Find determinant of a matrix A
Solution: Expand determinant on the first column:
= 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
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 formRule: 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
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):
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):
MatrixMatrix Definition. Main informationSystem of linear equations  matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix Add the comment 