JEE Main Matrices and Determinants 2027 — Properties, Adjoint, Inverse, Cramer’s Rule and 40 Practice MCQs

May 4, 2026 · · Blog

Last Updated: May 2026

JEE Main Matrices and Determinants 2027 consistently delivers 3-4 questions across the two papers in every JEE Main session, making it one of the most reliably scoring chapters in Mathematics. JEE Advanced often picks the toughest matrix-rank/inverse problems for one of the integer-type questions. This guide covers the complete syllabus, key theorems, properties, shortcuts and 40 practice MCQs.

Quick Facts: JEE Matrices & Determinants 2027

Aspect	Detail
NCERT chapters	Class XII Maths Ch 3 (Matrices) & Ch 4 (Determinants)
JEE Main questions/year	3-4
JEE Advanced	1-2 (often integer/match-the-column)
Difficulty	Easy to Moderate (calculation-heavy)
High-yield zones	Inverse, rank, system of equations, adjoint properties

Matrix Definitions and Types

Order: m × n; total elements = mn
Square matrix: m = n
Diagonal matrix: a_ij = 0 for i ≠ j
Scalar: diagonal with all equal entries
Identity (I): diagonal with all 1’s
Symmetric: A^T = A
Skew-symmetric: A^T = −A (diagonal entries = 0)
Orthogonal: AA^T = I
Idempotent: A² = A
Nilpotent: A^k = 0 for some k
Involutory: A² = I

Determinant — Key Properties (Memorise These!)

|A| = |A^T|
Swapping two rows/columns flips sign.
If two rows/columns are identical → |A| = 0.
Multiplying any row by k → |A| becomes k|A|.
|kA| = kⁿ|A| (n = order).
|AB| = |A||B|.
|A⁻¹| = 1/|A| (provided |A| ≠ 0).
Triangular/diagonal matrix: |A| = product of diagonal entries.
|adj A| = |A|ⁿ⁻¹ for an n × n matrix.
|adj(adj A)| = |A|^(n−1)².

Adjoint and Inverse

Cofactor: C_ij = (−1)^i+jM_ij
adj A = transpose of the cofactor matrix.
A·(adj A) = (adj A)·A = |A|·I
A⁻¹ = (adj A) / |A| — provided |A| ≠ 0 (singular vs non-singular).
For 2 × 2 matrix [[a, b],[c, d]]: A⁻¹ = (1/(ad − bc)) × [[d, −b],[−c, a]]

System of Linear Equations

Case	\|A\|	(adj A)·B	Conclusion
Unique solution	≠ 0	—	Consistent
Infinite solutions	= 0	= 0	Consistent — dependent
No solution	= 0	≠ 0	Inconsistent

Cramer’s Rule

For AX = B with |A| ≠ 0: x_i = |A_i| / |A|, where A_i is A with i-th column replaced by B.

Rank of a Matrix (JEE Advanced)

Rank = order of the largest non-zero minor.
Echelon form rank = number of non-zero rows after row-reduction.
Rank ≤ min(m, n).
System is consistent iff rank(A) = rank(A|B).

Special Determinant Identities

Identity	Value
Vandermonde 3×3 (1,a,a²; 1,b,b²; 1,c,c²)	(a−b)(b−c)(c−a)
Skew-symmetric of odd order	0
Determinant of orthogonal matrix	±1

JEE-Style Trap Concepts

(AB)^T = B^TA^T — order matters.
(AB)⁻¹ = B⁻¹A⁻¹ — same caveat.
Matrix multiplication is NOT commutative; always check given conditions.
If A is symmetric and B is skew-symmetric, AB + BA is skew, AB − BA is symmetric.
Trace of a matrix = sum of diagonal elements; tr(AB) = tr(BA).

FAQ — JEE Matrices & Determinants 2027

Q1. What is the relationship between adj A and |A|?

A · (adj A) = (adj A) · A = |A| · I, and |adj A| = |A|ⁿ⁻¹ where n is the order of the square matrix.

Q2. When does a system AX = B have no solution?

When |A| = 0 and (adj A)·B ≠ 0 — the system is inconsistent. Equivalently, when rank(A) ≠ rank(A|B).

Q3. What is a Vandermonde determinant?

A determinant of the form |1, a, a²; 1, b, b²; 1, c, c²| = (a−b)(b−c)(c−a). This identity is heavily exploited in JEE Main symmetry-based questions.

Q4. Why is the determinant of an odd-order skew-symmetric matrix zero?

Because |A^T| = |A|, and for skew-symmetric matrices A^T = −A, giving |A| = (−1)ⁿ|A|. For odd n, this forces |A| = 0.

Q5. What is the difference between Cramer’s rule and matrix inverse method?

Both solve AX = B when |A| ≠ 0. Cramer’s rule computes each variable as a determinant ratio, useful for 2×2 or 3×3 systems. Matrix inversion is more general but computationally heavier.

Practice MCQs

[cg_quiz id=”jee-matrices-determinants-2027″]