JEE Main Maths · Matrices & Determinants PYQ
JEE Main Matrices & Determinants PYQ (2002–2025)
Matrices & Determinants is a P1 Algebra chapter — ~6% JEE Main weightage with 2 questions per session routinely. Question templates are highly repetitive: determinant evaluation, matrix inverse, system of equations. Mastery here translates directly to mock scores.
Matrices & Determinants PYQs from 2002 to 2025, tagged by technique (row operations, expansion, inverse, Cramer's rule). Every solution shows step-by-step determinant manipulation with KaTeX.
Matrices & Determinants at a Glance
Key Sub-Topics & What's Tested
Matrix Algebra
Addition, subtraction, scalar multiplication, matrix multiplication (non-commutative), transpose, adjoint.
Determinants — Properties
Row/column operations, determinant of product = product of determinants, zero-row/column gives zero determinant.
Determinant Expansion
Cofactor expansion along any row or column, minors and cofactors, determinants of 2×2 and 3×3 matrices.
Inverse of a Matrix
A⁻¹ = adj(A)/|A|, condition |A| ≠ 0, properties of inverse: (AB)⁻¹ = B⁻¹A⁻¹, inverse of transpose = transpose of inverse.
System of Linear Equations
Using matrix inverse method X = A⁻¹B, Cramer's rule, consistency conditions (unique, infinite, no solution).
Special Matrices
Symmetric (Aᵀ = A), skew-symmetric (Aᵀ = -A), orthogonal (AAᵀ = I), idempotent (A² = A), nilpotent (Aⁿ = 0).
Determinant Applications
Area of triangle from coordinates, collinearity condition (det = 0), equation of line through two points.
Rank of a Matrix
Rank via row-reduction to echelon form, consistency of system based on rank of A vs rank of [A|B].
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Determinant Evaluation | 25% | Find determinant of |1 2 3; 4 5 6; 7 8 9| using row operations. |
| Matrix Inverse Computation | 20% | Find A⁻¹ for given 2×2 or 3×3 matrix using adjoint method. |
| System of Equations (Consistency) | 20% | For what values of k is the system consistent? Determine using determinant = 0 condition. |
| Matrix Properties / Special Types | 15% | If A is 3×3 symmetric and B is skew-symmetric, prove AB + BA is symmetric. |
| Determinant with Unknown | 10% | Find value of x such that |x 1 1; 1 x 1; 1 1 x| = 0. |
| Cramer's Rule Application | 10% | Solve 3-variable system using Cramer's rule; find one specific variable. |
How to Solve Matrices & Determinants PYQs
- 1Use row operations before expansion. Row reducing to create a zero column or row before expansion cuts determinant calculation time by 60%.
- 2For 3×3 determinants, cofactor expansion on the row/column with most zeros. Reduces arithmetic dramatically vs brute-force expansion.
- 3Check |A| ≠ 0 before computing inverse. Singular matrix has no inverse. Wasted computation if you skip this check.
- 4For system consistency, apply rank criterion. Rank(A) = Rank([A|B]) = n → unique solution. = r < n → infinite. Rank(A) < Rank([A|B]) → no solution.
- 5Cramer's rule for specific variable extraction. x_i = D_i / D where D is coefficient determinant, D_i replaces i-th column with RHS vector.
Common Mistakes That Cost Marks
- Matrix multiplication non-commutativity. AB ≠ BA in general. Don't assume they commute just because they're matrices.
- Sign errors in cofactor expansion. Sign of minor M_ij is (-1)^(i+j). Forgetting the sign gives wrong determinant.
- Using |A| for adjoint instead of A. adj(A) is a matrix of signed cofactors (transposed). |A| is a scalar. They're not interchangeable.
- Incorrect transpose of product. (AB)ᵀ = BᵀAᵀ (order reverses). Not AᵀBᵀ.
- Dividing matrix by matrix. Matrix division doesn't exist. To "divide" by matrix, multiply by inverse (AB = C → A = CB⁻¹, not C/B).
Related JEE Main Practice
Frequently asked questions
How many Matrices & Determinants PYQs should I solve?
Target 70–90 PYQs across 2010–2025. Given the high template-repeat rate (determinant evaluation, inverse, consistency), focused practice quickly reaches strong accuracy.
Is Matrices harder than Determinants?
Determinants are more pattern-based (expand along row/column with zeros). Matrices require more non-commutative algebra care. Both are equally tested — master both together.
Do I need Cramer's rule or matrix inverse method?
Both. Cramer's is faster for a single variable. Inverse method for all variables. PYQs test both — know when each is efficient.
What's the most-tested Matrices & Determinants pattern?
Determinant with unknown parameter k — find k such that det = 0. Roughly 3–4 variants per 10-year window.
Is rank of a matrix tested in JEE Main?
Rarely directly, but conceptually via system of equations (unique vs infinite vs no solution). Know the rank criterion — it unlocks many consistency questions.
How does Matrices & Determinants connect to 3D Geometry?
3×3 determinants are used for volume of parallelepiped, area of triangle in 3D, equation of plane through three points. Strong matrix fluency helps 3D Geometry PYQs.
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