Definition of Cofactors 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
Properties of Determinant
= bei +bfh +ceh - ceh – bei - bfh = 0
Expansion along the columns Expansion along the 1st column What should be the value of bA11 + eA21 + hA31? e h b = 0 Similarly, aA21 + bA22 + cA23 = 0.
Why?
Expansion along the columns Expansion along the 1st column What should be the value? How about Ans: k3detA
What is the value of = 0
If Then what is the value of = ? Ans: 0
Applications = (a + a’)A11 + (d + d’)A21 + (g + g’)A31 = (aA11 + dA21 + gA31) + (a’A11 + d’A21 + g’A31) Why?
Why?
Examples: = 80
= -67
The End.