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. 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.

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

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

12. 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

13. 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

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