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

 

For a 2 × 2 determinant

For

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

 

Minor will be

๐‘€ 11 ,  ๐‘€ 12 ,  ๐‘€ 21 ,  ๐‘€ 22

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And cofactors will be

๐ด 11 ,  ๐ด 12 ,  ๐ด 21 ,  ๐ด 22

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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

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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?

 

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  1. Chapter 4 Class 12 Determinants
  2. Concept wise

Transcript

For a 2 ร— 2 determinant For โˆ† = |โ– 8(3&2@1&4)| Minor will be ๐‘€_11, ๐‘€_12, ๐‘€_21, ๐‘€_22 M11 = |โ– 8(3&2@1&4)| = 4 M12 = |โ– 8(3&2@1&4)|= 1 M21 = |โ– 8(3&2@1&4)|= 2 M22 = |โ– 8(3&2@1&4)|= 3 And cofactors will be ๐ด_11, ๐ด_12, ๐ด_21, ๐ด_22 ๐ด_11 = ใ€–(โˆ’1)ใ€—^(1 + 1) ๐‘€_11 = ใ€–(โˆ’1)ใ€—^2 ๐‘€_11 = ๐‘€_11 = 4 ๐ด_12 = ใ€–(โˆ’1)ใ€—^(1 +2) ๐‘€_12 = ใ€–(โˆ’1)ใ€—^3 ๐‘€_12 = โˆ’1 ร— ๐‘€_12 = โˆ’1 ร— 1 = โˆ’1 ๐ด_21 = ใ€–(โˆ’1)ใ€—^(2 + 1) ๐‘€_21 = ใ€–(โˆ’1)ใ€—^3 ๐‘€_21 = โˆ’1 ร— ๐‘€_11 = โˆ’1 ร— 2 = โˆ’2 ๐ด_22 = ใ€–(โˆ’1)ใ€—^(2 +2) ๐‘€_22 = ใ€–(โˆ’1)ใ€—^4 ๐‘€_22 = 1 ร— ๐‘€_22 = 3 For a 3 ร— 3 matrix For โˆ† = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| We have elements, ๐‘Ž_11 = 9 ๐‘Ž_12 = 2 ๐‘Ž_13 = 1 ๐‘Ž_21 = 5 ๐‘Ž_22 = โˆ’1 ๐‘Ž_23 = 6 ๐‘Ž_31 = 1 ๐‘Ž_32 = 6 ๐‘Ž_33 = โˆ’2 . Minor will be ๐‘€_11, ๐‘€_12, ๐‘€_13, ๐‘€_21, ๐‘€_22, ๐‘€_23, ๐‘€_31, ๐‘€_32, ๐‘€_33 M11 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(โˆ’1&6@0&โˆ’2)| = (โˆ’1) ร— (โˆ’2) โˆ’ 0 ร— 6 = 2 โˆ’ 0 = 2 M12 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(5&6@4&โˆ’2)| = 5 ร— (โˆ’2) โˆ’ 4 ร— 6 = โˆ’10 โˆ’ 24 = โˆ’34 M13 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(5&โˆ’1@4&0)| = 5 ร— 0 โˆ’ 4 ร— (โˆ’1) = 0 + 4 = 4 M21 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(2&1@0&โˆ’2)| = 2 ร— (โˆ’2) โˆ’ 0 ร— 1 = โˆ’4 โˆ’ 0 = โˆ’4 M22 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(9&1@4&โˆ’2)| = 9 ร— (โˆ’2) โˆ’ 4 ร— 1 = โˆ’18 โˆ’ 4 = โˆ’ 22 M23 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(9&2@4&0)| = 9 ร— 0 โˆ’ 4 ร— 2 = 0 โˆ’ 8 = โˆ’ 8 M31 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(2&1@โˆ’1&6)| = 2 ร— 6 โˆ’ (โˆ’1) ร— 1 = 12 + 1 = 13 M32 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(9&1@5&6)| = 9 ร— 6 โˆ’ 5 ร— 1 = 54 โˆ’ 5 = 49 M33 = |โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)| = |โ– 8(9&2@5&โˆ’1)| = 9 ร— (โˆ’1) โˆ’ 5 ร— 2 = โˆ’9 โˆ’ 10 = โˆ’ 19 And, cofactors will be ๐ด_11 = ใ€–(โˆ’1)ใ€—^(1 + 1) ๐‘€_11 = ๐‘€_11 = 2 ๐ด_12 = ใ€–(โˆ’1)ใ€—^(1 + 2) ๐‘€_12 = โˆ’1 ร— ๐‘€_12 = โˆ’1 ร— โˆ’34 = 34 ๐ด_13 = ใ€–(โˆ’1)ใ€—^(1 + 3) ๐‘€_13 = ๐‘€_13 = 4 ๐ด_21 = ใ€–(โˆ’1)ใ€—^(2 + 1) ๐‘€_21 = โˆ’1 ร— ๐‘€_21 = โˆ’1 ร— โˆ’4 = 4 ๐ด_22 = ใ€–(โˆ’1)ใ€—^(2 + 2) ๐‘€_22 = ๐‘€_22 = โˆ’22 ๐ด_23 = ใ€–(โˆ’1)ใ€—^(2 +3) ๐‘€_23 = โˆ’1 ร— ๐‘€_23 = โˆ’1 ร— โˆ’8 = 8 ๐ด_31 = ใ€–(โˆ’1)ใ€—^(3 + 1) ๐‘€_31 = ๐‘€_31 = 13 ๐ด_32 = ใ€–(โˆ’1)ใ€—^(3 + 2) ๐‘€_32 = โˆ’1 ร— ๐‘€_32 = โˆ’1 ร— 49 = โˆ’49 ๐ด_33 = ใ€–(โˆ’1)ใ€—^(3 +3) ๐‘€_33 = ๐‘€_33 = โˆ’19 Note : We can also calculate cofactors without calculating minors If i + j is odd, Aij = โˆ’1 ร— Mij If i + j is even, Aij = Mij But, why use cofactor? Letโ€™s take a general determinant, โˆ† = |โ– 8(๐‘Ž_11&๐‘Ž_12&๐‘Ž_13@๐‘Ž_21&๐‘Ž_22&๐‘Ž_23@๐‘Ž_31&๐‘Ž_32&๐‘Ž_33 )| โˆ† = ๐‘Ž_11 |โ– 8(๐‘Ž_22&๐‘Ž_23@๐‘Ž_32&๐‘Ž_33 )| โˆ’ ๐‘Ž_12 |โ– 8(๐‘Ž_21&๐‘Ž_23@๐‘Ž_31&๐‘Ž_33 )| โˆ’ ๐‘Ž_13 |โ– 8(๐‘Ž_21&๐‘Ž_22@๐‘Ž_31&๐‘Ž_32 )| We can also write it as โˆ† = ๐‘Ž_11 ร— ใ€–(โˆ’1)ใ€—^(1+1) |โ– 8(๐‘Ž_22&๐‘Ž_23@๐‘Ž_32&๐‘Ž_33 )| + ๐‘Ž_12 ร— ใ€–(โˆ’1)ใ€—^(1+2) |โ– 8(๐‘Ž_22&๐‘Ž_23@๐‘Ž_32&๐‘Ž_33 )| + ๐‘Ž_13 ร— ใ€–(โˆ’1)ใ€—^(1+3) |โ– 8(๐‘Ž_22&๐‘Ž_23@๐‘Ž_32&๐‘Ž_33 )| โˆ† = ๐‘Ž_11 ๐ด_11 + ๐‘Ž_12 ๐ด_12 + ๐‘Ž_13 ๐ด_13

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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.