Matrices & Determinants (Class 12 Maths): Key Rules + Solved Questions
The must-know operations, properties of determinants, and how to solve linear equations by the matrix method — with worked examples.
Why it scores: Matrices and determinants are among the most predictable, high-scoring Class 12 topics. Master a few rules and most questions become mechanical.
Core rules to remember
- Matrix multiplication is not commutative: AB ≠ BA in general.
- (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹.
- A matrix is invertible only if its determinant ≠ 0.
- A⁻¹ = adj(A)/|A|.
Properties of determinants
| Property | Effect |
|---|---|
| Swapping two rows/columns | Sign of determinant changes |
| Two identical rows/columns | Determinant = 0 |
| Multiply a row by k | Determinant multiplied by k |
| |AB| | = |A| × |B| |
Worked example — solving equations
Q. Solve 2x + 3y = 8, x + 2y = 5 using the matrix method.
Write AX = B with A = [[2,3],[1,2]], so |A| = (2)(2) − (3)(1) = 1 ≠ 0. Then X = A⁻¹B. Here A⁻¹ = [[2,−3],[−1,2]], giving x = (2)(8)+(−3)(5) = 1 and y = (−1)(8)+(2)(5) = 2. So x = 1, y = 2.
Exam tips
- Always check |A| before finding A⁻¹.
- Use row operations to simplify large determinants.
- Watch signs in the cofactor/adjoint steps.
FAQs
Is the matrix method or Cramer's rule better for boards?
Both are accepted; the matrix (inverse) method is versatile and widely expected. Follow the method your question specifies.
How much weight do matrices carry?
Matrices and determinants together form a reliable scoring block. Confirm the exact split from the current syllabus.
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