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Determinants

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

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Determinants

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  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.

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