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 = matrix49

If we leave the row and column passing through aij (aij means the element belonging ith row and jth column) then we obtain a second order determinant which is minor of aij and is denoted by Mij. In general, minor Mij of an element aij is the determinant excluding ith row and jth 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 = matrix50

(ii) The minor of element a22 = M22 = matrix51

(iii) The minor of element a31 = M31 = matrix52 and so on.

        Cofactor of an element aij is defined as Cij = (-1)i+j Mij.

        Where, Cij = cofactor of aij.


Find the minors and cofactors of matrix53 along second column.


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  = ∑mi=1  aij .Cij, j = 1, 2, ...... m.

            = ∑mj=1  aij.Cij, 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.matrix55

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).


Expand the following determinant.

   Δ = matrix56 = ∑mi=1  aij cij


.·. Δ = a.matrix57  a(ek - f2) -b (dk - if) +c (dj - ie)

       = a M11 - b M12 + c M13 = a C11 + b C12 + c C13

Note : Though in the example, elements of first row and their cofactors are considered, the value of the determinant can be evaluated from any row and column.

Sarrus Rule: Sarrus give a rule for a determinant of order 3. Write down the three rows of a determinant and rewrite the first two rows. The three diagonals sloping down the right give the three positive terms and the three diagonals sloping down to the left give the three negative terms.


                                        .·. Δ = P - N.

IIT JEE study material is available online free of cost at askIITians.com. Study Physics, Chemistry and Mathematics at askIITians website and be a winner. We offer numerous live online courses as well for live online IIT JEE preparation - you do not need to travel anywhere any longer - just sit at your home and study for IIT JEE live online with askIITians.com


Contact askiitians experts to get answers to your queries by filling up the form given below:

We promise that your information will be our little secret. To know more please see our Privacy Policy
We promise that your information will be our little secret. To know more please see our Privacy Policy


Sign Up with Facebook

Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”

Related Resources
Operations on Determinants

Operations on Determinants Multiplication of two...

System of Linear Equations

System of Linear Equations System of Linear...

Algebra of Matrices

Algebra of Matrices Addition and Subtraction of...

Determinants Definitions Properties

Determinants Definitions & Properties...


Matrices: Definition and Classification DEFINITION...

System of Simultaneous Linear Equations

System of Simultaneous Linear Equations System of...

Elementary Operations of Matrix

Elementary Operations of a Matrix The following...

Solved Examples Part-1

IIT JEE Download IIT JEE Solved Examples on...

Special Matrices

Special Matrices Symmetric and Skew Matrices A...