Sec 3.6 Determinants. TH2: the invers of 2x2 matrix Recall from section 3.5 :

Apresentação em tema: "Sec 3.6 Determinants. TH2: the invers of 2x2 matrix Recall from section 3.5 :"— Transcrição da apresentação:

1 Sec 3.6 Determinants

2 TH2: the invers of 2x2 matrix Recall from section 3.5 :

3 Sec 3.6 Determinants Evaluate the determinant of 2x2 matrix How to compute the Higher-order determinants

4 Sec 3.6 Determinants Def: Minors Let A =[aij] be an nxn matrix. The ijth minor of A ( or the minor of aij) is the determinant Mij of the (n-1)x(n-1) submatrix after you delete the ith row and the jth column of A. Find

5 Sec 3.6 Determinants Def: Cofactors Let A =[aij] be an nxn matrix. The ijth cofactor of A ( or the cofactor of aij) is defined to be Find signs

6 Sec 3.6 Determinants 3x3 matrix signs Find det A

7 Sec 3.6 Determinants The cofactor expansion of det A along the first row of A Note:  3x3 determinant expressed in terms of three 2x2 determinants  4x4 determinant expressed in terms of four 3x3 determinants  5x5 determinant expressed in terms of five 4x4 determinants  nxn determinant expressed in terms of n determinants of size (n-1)x(n-1)

8 Sec 3.6 Determinants nxn matrix We multiply each element by its cofactor ( in the first row) Also we can choose any row or column Th1 : the det of an nxn matrix can be obtained by expansion along any row or column. i-th row j-th column

9 Row and Column Properties Prop 1: interchanging two rows (or columns)

10 Row and Column Properties Prop 2: two rows (or columns) are identical

11 Row and Column Properties Prop 3: (k) i-th row + j-th row (k) i-th col + j-th col

12 Row and Column Properties Prop 4: (k) i-th row (k) i-th col

13 Row and Column Properties Prop 5: i-th row B = i-th row A1 + i-th row A2 Prop 5: i-th col B = i-th col A1 + i-th col A2

14 Row and Column Properties Prop 6: det( triangular ) = product of diagonal Zeros below main diagonalZeros above main diagonal Either upper or lower

15 Row and Column Properties

16 Transpose Prop 6: det( matrix ) = det( transpose)

17 Transpose

18 Determinant and invertibility THM 2: The nxn matrix A is invertible detA = 0

19 Theorem7:(p193) row equivalentnonsingular Ax = b Every n-vector b has unique sol Ax = b Every n-vector b is consistent Ax = 0 The system has only the trivial sol is a product of elementary matrices

20 Determinant and inevitability THM 2: det ( A B ) = det A * det B Note: Proof: Example: compute

21 Solve the system Cramer’s Rule (solve linear system)

22 Sec 3.6 Determinants Solve the system Cramer’s Rule (solve linear system) Solve the system

23 Use cramer’s rule to solve the system Cramer’s Rule (solve linear system)

24 Adjoint matrix Def: Cofactor matrix Let A =[aij] be an nxn matrix. The cofactor matrix = [Aij] Find the cofactor matrix signs Def: Adjoint matrix of A Find the adjoint matrix

25 Another method to find the inverse Thm2: The inverse of A Find the inverse of A How to find the inverse of a matrix

26 Computational Efficiency The amount of labor required to compute a numerical calculation is measured by the number of arithmetical operations it involves Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion 2x2: 2 multiplications 3x3: three 2x2 determinants  3x2= 6 multiplications 4x4: four 3x3 determinants  4x3x2= 24 multiplications 5x5: four 3x3 determinants  4x3x2= 24 multiplications - - - - - - - - - - - - - - - - - - - - - - - - - - - - nxn: n (n-1)x(n-1) determinants  nx…x3x2= n! multiplications

27 Computational Efficiency Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion nxn: determinants  requires n! multiplications a typical 1998 desktop computer, using MATLAB and performing only 40 million operations per second To evaluate a determinant of a 15x15 matrix using cofactor expansion  requires a supercomputer capable of a billion operations per seconds To evaluate a detrminant of a 25x25 matrix using cofactor expansion  requires