Co-factor of Matrices

Each element which is associated with a 2*2 determinant then the values of that determinant are called cofactors. The cofactor is defined the signed minor. An (i,j) cofactor is computed by multiplying (i,j) minor by  and is denoted by . The formula to find cofactor =  where  denotes the minor of  row and  column of a matrix.

Co-factor of 2×2 order matrix

Let A be a square matrix. By cofactor  of an element  of A, we mean minor of  with a positive or negative sign depending on i and j. For a 2*2 matrix, negative sign is to be given the minor element  and  = 

Example 1: Consider the matrix 

Solution: The minor of 5 is 2 and Cofactor 5 is 2 (sign unchanged)

The minor of -1 is 2 and Cofactor -1 is -2 (sign changed)

The minor of 2 is -1 and Cofactor -1 is +1  (sign changed)

The minor of 2 is 5 and Cofactor 2 is 5 (sign unchanged)

Co- factor of 

Example 2: Consider the matrix 

Solution: The minor of 5 is 0 and Cofactor 5 is 0 (sign unchanged)

The minor of -3 is -2 and Cofactor -3 is +2 (sign changed)

The minor of -2 is -3 and Cofactor -2 is +3 (sign changed)

The minor of 0 is 5 and Cofactor 0 is 5 (sign unchanged)

Co- factor of 

Co-factor of 3×3 order matrix

For a 3*3 matrix, negative sign is to given to minor of element : 

Example 3: Consider the matrix 

Solution: Minor of 2 is 7 and Cofactor is 7.

Minor of -3 is 18 and Cofactor is -18 (sign changed)

Minor of -1 is 30 and Cofactor are 30.

Minor of 6 is 1 and Cofactor is -1 (sign changed)

Minor of 4 is 6 and Cofactor are 6.

Minor of 1 is 10 and Cofactor is -10 (sign changed)

Minor of 0 is 1 and Cofactor are 1.

Minor of 6 is 8 and Cofactor is -8 (sign changed)

Minor of 3 is 26 and Cofactor is 26

Example 4: Consider the matrix 

Solution: Minor of 3 is -26 and Cofactor is -26.

Minor of -2 is 18 and Cofactor is -8 (sign changed)

Minor of -1 is 12 and Cofactor is 12.

Minor of 2 is -2 and Cofactor is -2 (sign changed)

Minor of 1 is 12 and Cofactor are 12.

Minor of 5 is 18 and Cofactor is -18 (sign changed)

Minor of 0 is -9 and Cofactor are -9.

Minor of 6 is 17 and Cofactor is -17 (sign changed)

Minor of 4 is 7 and Cofactor are 7.

Exercise

1. Find the co-factors of the matrix .
2. Find the co-factors of matrix .
3. Find the co-factors of matrix.
4. Find the co-factors of matrix .
5. Find the co-factors of the matrix .