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4.Minor & Cofactor
Class 12 • Mathematics • NCERT
Minor & Cofactor
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This lesson explains the concepts of minor and cofactor of an element of a determinant as prescribed by NCERT Class 12. These ideas are essential for expanding determinants and finding inverse of matrices.
Lesson Objectives
- Understand the concept of a minor.
- Define cofactor and its sign.
- Use minors and cofactors in determinant expansion.
- Prepare for NCERT exam questions.
1. Minor of an Element
The minor of an element aᵢⱼ of a determinant is the determinant obtained by deleting the i-th row and j-th column containing that element.
Minor of aᵢⱼ is denoted by Mᵢⱼ
2. Example of Minor
For determinant:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
Minor of element 2 (a₁₂) is:
| 4 6 |
| 7 9 |
3. Cofactor of an Element
The cofactor of an element aᵢⱼ is defined as:
Cᵢⱼ = (−1)i+j Mᵢⱼ
The sign of the cofactor depends on the position (i + j).
4. Sign Pattern of Cofactors
| + − + |
| − + − |
| + − + |
| − + − |
| + − + |
This pattern is very important for exams.
5. Expansion Using Cofactors
The value of a determinant can be obtained by expanding along any row or column using cofactors.
|A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃
Practice Questions (NCERT)
- Define the minor of an element.
- Define the cofactor of an element.
- Write the formula for cofactor Cᵢⱼ.
- What is the sign of C₂₃?
- Find the minor of a₁₁ in a 3×3 determinant.
- State the sign pattern of cofactors.
- Can determinant be expanded along any row?
- Is minor a determinant or an element?
- What does (−1)i+j represent?
- Is this topic part of NCERT Class 12 syllabus?
✅ Show Answer Key
- Determinant after deleting row and column
- Cofactor = (−1)i+j × minor
- Cᵢⱼ = (−1)i+j Mᵢⱼ
- Negative
- Determinant of remaining 2×2 matrix
- + − + / − + − / + − +
- Yes
- Determinant
- Sign of cofactor
- Yes
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