Determinants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262

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1 Determinants Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations 1 / 24

2 Outline 1 The definition of the determinant 2 Properties of determinants 3 Cofactor expansion Samy T. Determinants Differential equations 2 / 24

3 Outline 1 The definition of the determinant 2 Properties of determinants 3 Cofactor expansion Samy T. Determinants Differential equations 3 / 24

4 Particular cases 1 1 matrix: A = [a 11 ] = det(a) = a matrix: [ ] a11 a A = 12 a 21 a 22 = det(a) = a 11 a 22 a 12 a matrix: Samy T. Determinants Differential equations 4 / 24

5 Remarks Generalization: The determinant is defined for any n n matrix Combinatorics involved Motivation: In general det(a) 0 A is invertible Notation: det(a) A Samy T. Determinants Differential equations 5 / 24

6 Examples 2 2 matrix: = matrix: = Samy T. Determinants Differential equations 6 / 24

7 Outline 1 The definition of the determinant 2 Properties of determinants 3 Cofactor expansion Samy T. Determinants Differential equations 7 / 24

8 Introduction Problem with determinants: For a n n, matrix, they require n! operations This is computationally too demanding Aim of this section: See properties in order to shorten computation time Samy T. Determinants Differential equations 8 / 24

9 Determinants of triangular matrices Theorem 1. Let A be an upper or lower triangular matrix. n size of A. Then det(a) = a 11 a 22 a nn = n a ii i=1 Example: = Samy T. Determinants Differential equations 9 / 24

10 Elementary row operations and determinants Effect of elementary row operations: If A is a n n matrix, then 1 Let B be the matrix obtained by permuting 2 rows of A. Then det(b) = det(a) 2 Let B obtained by multiplying 1 row of A by k R. Then det(b) = k det(a) 3 Let B obtained by adding k a row of A to a different row of A. Then det(b) = det(a) Samy T. Determinants Differential equations 10 / 24

11 Example of application 3 3 matrix: A 12 ( 2), A 13 ( 1) = M 2 ( 1), M 3 ( 1) = ( 1) 2 A 23 ( 3) = = Remark: This technique is really useful for n 4 Samy T. Determinants Differential equations 11 / 24

12 Further properties of determinants Some more properties: 4 We have det(a T ) = det(a) 5 If A has a column of 0 s, then det(a) = 0 6 If 2 rows or columns of A are the same, then det(a) = 0 7 For two matrices A and B, we have det(a B) = det(a) det(b) Samy T. Determinants Differential equations 12 / 24

13 Application of Property 4 Example: When further simplifications are available for columns = A 23 ( 5) = = Samy T. Determinants Differential equations 13 / 24

14 Outline 1 The definition of the determinant 2 Properties of determinants 3 Cofactor expansion Samy T. Determinants Differential equations 14 / 24

15 Introduction Aim: Reduce the order of a determinant by an expansion Vocabulary: First we have to introduce the notions of Minor Cofactor Samy T. Determinants Differential equations 15 / 24

16 Minors of a matrix Definition 2. Let A be a n n matrix. Then M ij = det(matrix obtained by deleting ith row and jth column of A) The quantity M ij is called minor of a ij. Example: A = = M 12 = 1 6 = Samy T. Determinants Differential equations 16 / 24

17 Cofactors of a matrix Definition 3. Let A be a n n matrix. Then C ij = ( 1) i+j M ij The quantity C ij is called cofactor of a ij. Example: A = = C 12 = M 12 = Remark: Alternate signs assignment for C ij Samy T. Determinants Differential equations 17 / 24

18 Cofactor expansion Theorem 4. Let A be a n n matrix. Then 1 One can expand the determinant along a row i: n det(a) = a ik C ik k=1 2 One can expand the determinant along a column j: n det(a) = a kj C kj k=1 Samy T. Determinants Differential equations 18 / 24

19 Example of application Rule: To simplify computations, choose row or column with 0 s Example: Here we expand along the 3rd row = Samy T. Determinants Differential equations 19 / 24

20 Adjoint matrix Definition 5. Let A be a n n matrix. Then Matrix of cofactors: Obtained by replacing each term of A by its cofactor Denoted by M C Adjoint matrix: Denoted by adj(a) and defined as adj(a) = M T C Samy T. Determinants Differential equations 20 / 24

21 The adjoint method Theorem 6. Let A be a n n matrix. Assume: det(a) 0. Then A 1 = 1 det(a) adj(a). Remark: Along the same lines we have A invertible det(a) 0 Samy T. Determinants Differential equations 21 / 24

22 Example Matrix: A = Cofactor and adjoint matrix: M C = 6 9 4, adj(a) = Inverse: det(a) = 55 and thus A 1 = Samy T. Determinants Differential equations 22 / 24

23 Cramer s rule Theorem 7. Consider a n n matrix A, a vector b and the system Ax = b. (1) For 1 k n set (b inserted at column k): a 11 a b 1... a 1n a B k = 21 a b 2... a 2n a n1 a n2... b n... a nn Then if det(a) 0 the solution of (1) is given by x k = det(b k) det(a) Samy T. Determinants Differential equations 23 / 24

24 Example System: 3x 1 +2x 2 x 3 = 4 x 1 +x 2 5x 3 = 3 2x 1 x 2 +4x 3 = 0 Determinants: det(a) = = 8, det(b 1 ) = = Solution: x 1 = 17 8 Samy T. Determinants Differential equations 24 / 24

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