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 Aij = (-1)i + j(2 x 2 determinant obtained by deleting the i-th row and the j-th column)

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)

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)

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

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

Expansion along the columns Expansion along the 1st column What should be the value of bA11 + eA21 + hA31? e h b C1 – C2 = 0 Similarly, aA21 + bA22 + cA23 = 0.

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

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

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

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

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

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

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