Determinants

1. Determinants Definition of Cofactors

2. Definition of Cofactors • Let M = • The cofactor of the i-th row and the j-th column is defined by • Aij = (-1)i + j(2 x 2 determinant obtained by deleting the i-th row and the j-th column)

3. Definition of Cofactors • Let M = • The cofactor of the i-th row and the j-th column is defined by • Aij = (-1)i + j(2 x 2 determinant obtained by deleting the i-th row and the j-th column)

4. Definition of Cofactors • Let M = • The cofactor of the i-th row and the j-th column is defined by • Aij = (-1)i + j(2 x 2 determinant obtained by deleting the i-th row and the j-th column)

5. Relation between Cofactors and Determinants • Let M = • det M = aei + bfg + cdh – ceg – afh – bdi Expansion along the 1st row

6. Expansion along the 2nd row • Let M = • det M = aei + bfg + cdh – ceg – afh – bdi Expansion along the 2nd row

7. Expansion along the columns Expansion along the 1st column

8. Properties of Determinant

9. = bei +bfh +ceh - ceh – bei - bfh = 0

10. b e b e h h = 0 Expansion along the columns Expansion along the 1st column • What should be the value of • bA11 + eA21 + hA31? • Similarly, aA21 + bA22 + cA23 = 0.

11. Why?

12. How about Expansion along the columns Expansion along the 1st column • What should be the value? Ans: k3detA

13. What is the value of = 0

14. If Then what is the value of = ? Ans: 0

15. Applications • = (a + a’)A11 + (d + d’)A21 + (g + g’)A31 • = (aA11 + dA21 + gA31) + (a’A11 + d’A21 + g’A31) Why?

16. Why?

17. Examples: = 80

18. = -67

19. The End.