JEE Main 2027 — Matrices and Determinants is one of the most scoring chapters in JEE Main Mathematics. It is straightforward in concept and yields 2–3 questions every year, predominantly from determinant properties, inverse of matrix, and system of linear equations. Master this chapter completely for guaranteed marks.
1. Introduction to Matrices
A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, it is called an m × n matrix (read as “m by n matrix”), and the order or size of the matrix is m × n.
The element in the i-th row and j-th column is denoted aij. The general matrix A of order m × n is written as A = [aij]m×n.
2. Types of Matrices
| Type | Definition | Example |
|---|---|---|
| Row Matrix | Only 1 row (order 1 × n) | [1, 2, 3] |
| Column Matrix | Only 1 column (order m × 1) | [[1],[2],[3]] |
| Square Matrix | Equal rows and columns (m = n) | [[1,2],[3,4]] |
| Null (Zero) Matrix | All elements are zero | [[0,0],[0,0]] |
| Identity Matrix (I) | Diagonal elements = 1, rest = 0 | [[1,0],[0,1]] |
| Diagonal Matrix | Non-diagonal elements = 0 | [[5,0],[0,3]] |
| Symmetric Matrix | A = AT (i.e., aij = aji) | [[1,2],[2,3]] |
| Skew-Symmetric | A = −AT (i.e., aij = −aji); diagonal = 0 | [[0,2],[-2,0]] |
| Idempotent Matrix | A² = A | [[1,0],[0,0]] |
| Nilpotent Matrix | Ak = 0 for some positive integer k | [[0,1],[0,0]] |
3. Matrix Operations
Addition and Subtraction
Two matrices can be added or subtracted only if they are of the same order. Addition is performed element-wise: (A + B)ij = aij + bij.
Properties: A + B = B + A (commutative); (A + B) + C = A + (B + C) (associative); A + O = A (O = null matrix).
Matrix Multiplication
Matrix AB is defined only when the number of columns of A equals the number of rows of B. If A is m × p and B is p × n, then AB is m × n.
(AB)ij = Σ aik · bkj (sum over k from 1 to p)
Important: Matrix multiplication is generally NOT commutative: AB ≠ BA. But it IS associative: (AB)C = A(BC). Also AI = IA = A.
Transpose of a Matrix
The transpose AT of matrix A is obtained by interchanging rows and columns: (AT)ij = aji.
Important properties: (AT)T = A; (A + B)T = AT + BT; (AB)T = BTAT (reversal rule — frequently tested in JEE).
4. Determinants
Every square matrix has a scalar value associated with it called its determinant. For a 2×2 matrix A = [[a,b],[c,d]]: det(A) = ad − bc.
Expansion of a 3×3 Determinant
For a 3×3 matrix, the determinant is expanded along any row or column using cofactors:
det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
Properties of Determinants — JEE High Yield
| Property | Statement |
|---|---|
| P1 — Transpose | det(A) = det(AT) |
| P2 — Row/Column swap | Interchanging two rows/columns reverses sign of determinant |
| P3 — Identical rows | If two rows/columns are identical, det = 0 |
| P4 — Scalar multiple | If all elements of one row are multiplied by k, det is multiplied by k. For n×n matrix: det(kA) = kn·det(A) |
| P5 — Product | det(AB) = det(A)·det(B) |
| P6 — Zero row | If all elements of a row/column are zero, det = 0 |
| P7 — Proportional rows | If one row is proportional to another, det = 0 |
5. Inverse of a Matrix
The inverse of a square matrix A exists if and only if det(A) ≠ 0 (i.e., A is non-singular).
Adjoint Method
The adjoint of matrix A, written adj(A), is the transpose of the cofactor matrix of A.
For 2×2 matrix A = [[a,b],[c,d]]: adj(A) = [[d,−b],[−c,a]]
Then: A⁻¹ = adj(A) / det(A)
Cramer’s Rule
For a system of n linear equations in n unknowns AX = B, Cramer’s rule gives:
x₁ = D₁/D, x₂ = D₂/D, …, xₙ = Dₙ/D
Where D = det(A), and Dᵢ is the determinant obtained by replacing the i-th column of A with column matrix B.
6. System of Linear Equations
For the system AX = B:
- Consistent with unique solution: rank(A) = rank(A|B) = n (number of unknowns)
- Consistent with infinite solutions: rank(A) = rank(A|B) < n
- Inconsistent (no solution): rank(A) ≠ rank(A|B)
For homogeneous system AX = 0: Always consistent (trivial solution X = 0). Non-trivial solutions exist iff det(A) = 0.
7. JEE Main Pattern — Matrices and Determinants
JEE Main typically asks 2–3 questions from this chapter every year:
- Determinant calculation (3×3) using properties to simplify
- Finding inverse of a 2×2 or 3×3 matrix
- System of equations — consistency check
- Transpose rule (AB)ᵀ = BᵀAᵀ
- Properties like det(kA) = kn·det(A) for n×n matrix
- Idempotent, nilpotent, involutory matrix identification
Practice MCQs — Matrices and Determinants
Attempt these 10 JEE Main level MCQs to test your preparation:
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Frequently Asked Questions (FAQs)
Q1. Is matrix multiplication commutative?
No. Matrix multiplication is generally NOT commutative — AB ≠ BA in most cases. However, multiplication is associative: (AB)C = A(BC). Some special cases where AB = BA include: (1) when one of the matrices is an identity matrix; (2) when both matrices are diagonal matrices; (3) some scalar matrices. Always verify before assuming commutativity.
Q2. What is the difference between singular and non-singular matrices?
A square matrix is called singular if its determinant is zero (det(A) = 0). A singular matrix does NOT have an inverse. A non-singular matrix has a non-zero determinant and always has a unique inverse. In the context of linear equations, a singular coefficient matrix means either no solution or infinitely many solutions (never a unique solution).
Q3. What is meant by the rank of a matrix?
The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. It equals the number of non-zero rows in the row echelon form. For an n×n non-singular matrix, rank = n. Rank is crucial for determining the nature of solutions of a system of linear equations.
Q4. How many questions come from Matrices in JEE Main?
JEE Main typically features 2–3 questions from Matrices and Determinants. The chapter is highly predictable — key areas are determinant properties (especially simplification using row/column operations), inverse of matrix, Cramer’s rule, and system of linear equations. This chapter offers some of the most reliable scoring opportunities in JEE Main Mathematics.