1. Sec 3.6 Determinants
2x2 matrix
Evaluate the determinant of

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

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

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. nxn matrix Sec 3.6 Determinants
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 row

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

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

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

22. Adjoint matrix Def: Cofactor matrix Def: Adjoint matrix of A
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

23. Another method to find the inverse
Thm2: The inverse of A
Find the inverse of A

24. 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

25. 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 aonly 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

26. Monday
Quiz