1 / 14

Section 8.4: Determinants

Section 8.4: 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 =

ima-hurst
Download Presentation

Section 8.4: Determinants

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 8.4: 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. 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? C1 – C2 • Similarly, aA21 + bA22 + cA23 = 0.

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

  9. Adjoint Matrix • Let M = • The adjoint matrix of M is defined by • adj M =

  10. det M Expansion along the first row The product of M and adj M • M(adj M) =

  11. det M Expansion along the second row The product of M and adj M • M(adj M) = det M det M

  12. 0 dA21 + eA22 + fA23 = det M, but aA21 + bA22 + cA23 = 0. The product of M and adj M • M(adj M) = det M det M det M

  13. 0 0 0 0 0 0 gA31 + hA32 + iA33 = det M, but aA31 + bA32 + cA33 = 0. The product of M and adj M • M(adj M) = det M 0 = (det M)I det M det M

  14. Conclusion • Let M be a square matrix. • Then M(adj M) = (adj M)M = (det M)I. • If det M  0, then

More Related