Minors and Co-Factors

The minor of an element of a determinant is again a determinant (of lesser order) formed by excluding the row and column of the element. For example take the following determinant

        Δ =

If we leave the row and column passing through a_ij (a_ij means the element belonging i^th row and j^th column) then we obtain a second order determinant which is minor of a_ij and is denoted by M_ij. In general, minor M_ij of an element a_ij is the determinant excluding i^th row and j^th column. Thus we have 9 minors corresponding to 9 elements of above determinant Δ. Here we are illustrating some minors of the determinant Δ.

(i) The minor of element a₁₁ = M₁₁ =

(ii) The minor of element a₂₂ = M₂₂ =

(iii) The minor of element a₃₁ = M₃₁ = and so on.

        Cofactor of an element a_ij is defined as C_ij = (-1)^i+j Mij.

        Where, C_ij = cofactor of a_ij.

Illustration:

Find the minors and cofactors of along second column.

Solution:

Minors along second column i.e. elements 2, 5 and 8 are

And cofactors of the corresponding elements are

        C₁₂ = (-1)¹⁺² (-6) = 6

        C₂₂ = (-1)²⁺² (-12) = -12

        C₃₂ = (-1)³⁺² (-6) = 6 respectively.

Evaluation of a Determinant

The determinant of order m can be evaluated as

        Δ  = ∑^m_i=1  a_ij .C_ij, j = 1, 2, ...... m.

            = ∑^m_j=1  a_ij.C_ij, i = 1, 2, ...... m.

i.e. the determinant can be evaluated by multiplying the elements of a single row or a column with their respective co-factors and then adding them.

For e.g.

Cofactor of a = d(-1)¹⁺¹ = d

Cofactor of b = c(-1)²⁺¹ = -c

So, the value of the determinant is (ad - bc).