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Understanding Cofactors and Their Role in Matrix Determinants

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This section delves into the definition and significance of cofactors in matrix operations. A cofactor for an element in the i-th row and j-th column is defined as Aij = (-1)^(i+j) times the determinant of the matrix obtained by deleting the respective row and column. The relationship between cofactors and determinants is explored through expansion formulas along different rows and columns. Additionally, it discusses the adjoint matrix and the properties of products involving the original matrix and its adjoint, emphasizing the role of determinant values.

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Understanding Cofactors and Their Role in Matrix Determinants

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  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

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