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3.Determinants – Definition & Properties
Class 12 • Mathematics • NCERT
Determinants – Definition & Properties
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This lesson introduces determinants as prescribed by NCERT Class 12. Students learn the definition of determinants and their important properties, which are essential for solving matrix equations and higher-level applications.
Lesson Objectives
- Understand the meaning of determinants.
- Evaluate determinants of order 2 and 3.
- Learn important properties of determinants.
- Prepare for NCERT exam questions.
1. Determinant
The determinant is a scalar value associated with a square matrix. It is denoted by |A| or det(A).
Determinant is defined only for square matrices.
2. Determinant of a 2×2 Matrix
For a matrix A =
| a b |
| c d |
| c d |
|A| = ad − bc
3. Determinant of a 3×3 Matrix
For a 3×3 matrix, determinant can be evaluated using expansion along any row or column.
| a b c |
| d e f |
| g h i |
| d e f |
| g h i |
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
4. Important Properties of Determinants
• Interchanging two rows (or columns) changes the sign of determinant
• If two rows (or columns) are identical, determinant = 0
• If one row is a multiple of another, determinant = 0
• Determinant of identity matrix = 1
• If two rows (or columns) are identical, determinant = 0
• If one row is a multiple of another, determinant = 0
• Determinant of identity matrix = 1
5. Important NCERT Notes
• Determinant of zero matrix = 0
• Determinant helps check consistency of equations
• Properties reduce calculation effort
• Determinant helps check consistency of equations
• Properties reduce calculation effort
Practice Questions (NCERT)
- Define determinant.
- For which matrices is determinant defined?
- Find the determinant of |2 3; 1 4|.
- State the effect of interchanging two rows of a determinant.
- What is the determinant of an identity matrix?
- If two rows of a determinant are equal, what is its value?
- Find the determinant of a zero matrix.
- Is determinant a scalar or matrix?
- Can a non-square matrix have a determinant?
- Is this topic part of NCERT Class 12 syllabus?
✅ Show Answer Key
- A scalar associated with a square matrix
- Square matrices
- 5
- Sign changes
- 1
- 0
- 0
- Scalar
- No
- Yes
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