Let’s look at what are minors & cofactor of a 2 × 2 & a 3 × 3 determinant

 

For a 2 × 2 determinant

For

16.jpg

We have elements,
    π‘Ž 11 = 3
    π‘Ž 12 = 2
    π‘Ž 21 = 1
    π‘Ž 22 = 4

 

Minor will be

𝑀 11 ,  𝑀 12 ,  𝑀 21 ,  𝑀 22

18.jpg

And cofactors will be

𝐴 11 ,  𝐴 12 ,  𝐴 21 ,  𝐴 22

19.jpg

 

For a 3 × 3 matrix

20.jpg

 

Minor will be

M 11 , M 12 , M 13 , M 21 , M 22 , M 23 , M 31 , M 32 , M 33

21.jpg

22.jpg

Note : We can also calculate cofactors without calculating minors

If i + j is odd,

A ij = −1 × M ij

If i + j is even,

A ij = M ij

But, why use cofactor?

 

23.jpg

 

  1. Chapter 4 Class 12 Determinants
  2. Concept wise

Transcript

For a 2 Γ— 2 determinant For βˆ† = |β– 8(3&2@1&4)| Minor will be 𝑀_11, 𝑀_12, 𝑀_21, 𝑀_22 M11 = |β– 8(3&2@1&4)| = 4 M12 = |β– 8(3&2@1&4)|= 1 M21 = |β– 8(3&2@1&4)|= 2 M22 = |β– 8(3&2@1&4)|= 3 And cofactors will be 𝐴_11, 𝐴_12, 𝐴_21, 𝐴_22 𝐴_11 = γ€–(βˆ’1)γ€—^(1 + 1) 𝑀_11 = γ€–(βˆ’1)γ€—^2 𝑀_11 = 𝑀_11 = 4 𝐴_12 = γ€–(βˆ’1)γ€—^(1 +2) 𝑀_12 = γ€–(βˆ’1)γ€—^3 𝑀_12 = βˆ’1 Γ— 𝑀_12 = βˆ’1 Γ— 1 = βˆ’1 𝐴_21 = γ€–(βˆ’1)γ€—^(2 + 1) 𝑀_21 = γ€–(βˆ’1)γ€—^3 𝑀_21 = βˆ’1 Γ— 𝑀_11 = βˆ’1 Γ— 2 = βˆ’2 𝐴_22 = γ€–(βˆ’1)γ€—^(2 +2) 𝑀_22 = γ€–(βˆ’1)γ€—^4 𝑀_22 = 1 Γ— 𝑀_22 = 3 For a 3 Γ— 3 matrix For βˆ† = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| We have elements, π‘Ž_11 = 9 π‘Ž_12 = 2 π‘Ž_13 = 1 π‘Ž_21 = 5 π‘Ž_22 = βˆ’1 π‘Ž_23 = 6 π‘Ž_31 = 1 π‘Ž_32 = 6 π‘Ž_33 = βˆ’2 . Minor will be 𝑀_11, 𝑀_12, 𝑀_13, 𝑀_21, 𝑀_22, 𝑀_23, 𝑀_31, 𝑀_32, 𝑀_33 M11 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(βˆ’1&6@0&βˆ’2)| = (βˆ’1) Γ— (βˆ’2) βˆ’ 0 Γ— 6 = 2 βˆ’ 0 = 2 M12 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(5&6@4&βˆ’2)| = 5 Γ— (βˆ’2) βˆ’ 4 Γ— 6 = βˆ’10 βˆ’ 24 = βˆ’34 M13 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(5&βˆ’1@4&0)| = 5 Γ— 0 βˆ’ 4 Γ— (βˆ’1) = 0 + 4 = 4 M21 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(2&1@0&βˆ’2)| = 2 Γ— (βˆ’2) βˆ’ 0 Γ— 1 = βˆ’4 βˆ’ 0 = βˆ’4 M22 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(9&1@4&βˆ’2)| = 9 Γ— (βˆ’2) βˆ’ 4 Γ— 1 = βˆ’18 βˆ’ 4 = βˆ’ 22 M23 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(9&2@4&0)| = 9 Γ— 0 βˆ’ 4 Γ— 2 = 0 βˆ’ 8 = βˆ’ 8 M31 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(2&1@βˆ’1&6)| = 2 Γ— 6 βˆ’ (βˆ’1) Γ— 1 = 12 + 1 = 13 M32 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(9&1@5&6)| = 9 Γ— 6 βˆ’ 5 Γ— 1 = 54 βˆ’ 5 = 49 M33 = |β– 8(9&2&1@5&βˆ’1&6@4&0&βˆ’2)| = |β– 8(9&2@5&βˆ’1)| = 9 Γ— (βˆ’1) βˆ’ 5 Γ— 2 = βˆ’9 βˆ’ 10 = βˆ’ 19 And, cofactors will be 𝐴_11 = γ€–(βˆ’1)γ€—^(1 + 1) 𝑀_11 = 𝑀_11 = 2 𝐴_12 = γ€–(βˆ’1)γ€—^(1 + 2) 𝑀_12 = βˆ’1 Γ— 𝑀_12 = βˆ’1 Γ— βˆ’34 = 34 𝐴_13 = γ€–(βˆ’1)γ€—^(1 + 3) 𝑀_13 = 𝑀_13 = 4 𝐴_21 = γ€–(βˆ’1)γ€—^(2 + 1) 𝑀_21 = βˆ’1 Γ— 𝑀_21 = βˆ’1 Γ— βˆ’4 = 4 𝐴_22 = γ€–(βˆ’1)γ€—^(2 + 2) 𝑀_22 = 𝑀_22 = βˆ’22 𝐴_23 = γ€–(βˆ’1)γ€—^(2 +3) 𝑀_23 = βˆ’1 Γ— 𝑀_23 = βˆ’1 Γ— βˆ’8 = 8 𝐴_31 = γ€–(βˆ’1)γ€—^(3 + 1) 𝑀_31 = 𝑀_31 = 13 𝐴_32 = γ€–(βˆ’1)γ€—^(3 + 2) 𝑀_32 = βˆ’1 Γ— 𝑀_32 = βˆ’1 Γ— 49 = βˆ’49 𝐴_33 = γ€–(βˆ’1)γ€—^(3 +3) 𝑀_33 = 𝑀_33 = βˆ’19 Note : We can also calculate cofactors without calculating minors If i + j is odd, Aij = βˆ’1 Γ— Mij If i + j is even, Aij = Mij But, why use cofactor? Let’s take a general determinant, βˆ† = |β– 8(π‘Ž_11&π‘Ž_12&π‘Ž_13@π‘Ž_21&π‘Ž_22&π‘Ž_23@π‘Ž_31&π‘Ž_32&π‘Ž_33 )| βˆ† = π‘Ž_11 |β– 8(π‘Ž_22&π‘Ž_23@π‘Ž_32&π‘Ž_33 )| βˆ’ π‘Ž_12 |β– 8(π‘Ž_21&π‘Ž_23@π‘Ž_31&π‘Ž_33 )| βˆ’ π‘Ž_13 |β– 8(π‘Ž_21&π‘Ž_22@π‘Ž_31&π‘Ž_32 )| We can also write it as βˆ† = π‘Ž_11 Γ— γ€–(βˆ’1)γ€—^(1+1) |β– 8(π‘Ž_22&π‘Ž_23@π‘Ž_32&π‘Ž_33 )| + π‘Ž_12 Γ— γ€–(βˆ’1)γ€—^(1+2) |β– 8(π‘Ž_22&π‘Ž_23@π‘Ž_32&π‘Ž_33 )| + π‘Ž_13 Γ— γ€–(βˆ’1)γ€—^(1+3) |β– 8(π‘Ž_22&π‘Ž_23@π‘Ž_32&π‘Ž_33 )| βˆ† = π‘Ž_11 𝐴_11 + π‘Ž_12 𝐴_12 + π‘Ž_13 𝐴_13

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.