Sec 3.6 Determinants

1. Sec 3.6 Determinants

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

3. Sec 3.6 Determinants 2x2 matrix Evaluate the determinant of 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. signs Sec 3.6 Determinants 3x3 matrix 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 Either upper or lower Zeros below main diagonal Zeros above main diagonal Prop 6: det( triangular ) = product of diagonal

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 equivalent nonsingular is a product of elementary matrices Every n-vector b The system Every n-vector b Ax = b Ax = 0 Ax = b has unique sol has only the trivial sol is consistent All statements are equivalent

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

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

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

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

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

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

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