Matrices and Determinants — JEE Main Mathematics Notes & Formulas

1. Chapter at a Glance

A matrix is a rectangular arrangement of numbers in rows and columns; a determinant is a single number attached to a square matrix. Together they give a compact way to solve systems of linear equations and to describe transformations. This is one of the highest-scoring chapters of JEE Main — usually 2 or 3 questions per shift — because questions are short and formula-based.

What JEE asks most

Properties of adjoint and inverse: |adj A|, adj(adj A), |kA|.
Consistency of a system of three equations (unique, none or infinitely many solutions).
Powers of special matrices (rotation matrices, nilpotent and idempotent matrices).
Symmetric and skew-symmetric matrices, and counting matrices with given properties.

2. Matrices — Types and Operations

A matrix of order m × n has m rows and n columns. The entry in the i-th row and j-th column is written a_ij.

Types of matrices

Type	Definition
Square	m = n
Diagonal	Square, all non-diagonal entries 0
Scalar	Diagonal with all diagonal entries equal
Identity I	Scalar matrix with diagonal entries 1
Upper / lower triangular	All entries below / above the diagonal are 0
Symmetric	Aᵀ = A (a_ij = a_ji)
Skew-symmetric	Aᵀ = −A (so every diagonal entry is 0)
Orthogonal	AAᵀ = AᵀA = I
Idempotent	A² = A
Nilpotent (index k)	A^k = O but A^k−1 ≠ O
Involutory	A² = I

Operations

Addition is defined only for matrices of the same order and is commutative.
Multiplication AB is defined when the number of columns of A equals the number of rows of B. In general AB ≠ BA, and AB = O does not mean A = O or B = O.
Transpose: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ (order reverses).
Every square matrix is the sum of a symmetric and a skew-symmetric matrix: A = ½(A + Aᵀ) + ½(A − Aᵀ).
Trace (sum of diagonal entries): tr(A + B) = tr A + tr B and tr(AB) = tr(BA).

3. Determinants and their Properties

For a 2 × 2 matrix, |A| = ad − bc. For a 3 × 3 matrix, expand along any row or column using cofactors C_ij = (−1)^i+j M_ij, where M_ij is the minor (determinant left after deleting row i and column j).

Geometric meaning: |det| of a 2 × 2 matrix is the area of the parallelogram formed by its rows (for 3 × 3, the volume of a parallelepiped).

Properties that save time

|Aᵀ| = |A|; interchanging two rows changes the sign; two identical (or proportional) rows make the determinant 0.
Multiplying one row by k multiplies the determinant by k, so for an n × n matrix |kA| = kⁿ|A|.
Adding a multiple of one row to another row does not change the determinant (the key tool for simplification).
|AB| = |A||B| and |Aⁿ| = |A|ⁿ.
The determinant of a triangular or diagonal matrix is the product of its diagonal entries.
A skew-symmetric matrix of odd order has determinant 0.
Vandermonde: |1 1 1; a b c; a² b² c²| = (a − b)(b − c)(c − a).

Area of a triangle

Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

Three points are collinear exactly when this determinant is 0.

4. Adjoint and Inverse

The adjoint adj A is the transpose of the matrix of cofactors. The key identity is

A (adj A) = (adj A) A = |A| I

So if |A| ≠ 0 (A is non-singular), the inverse is A⁻¹ = (adj A)/|A|. For a 2 × 2 matrix [a b; c d], adj A = [d −b; −c a] — swap the diagonal, change the sign of the other two.

Adjoint results for an n × n matrix

|adj A| = |A|ⁿ⁻¹
adj(adj A) = |A|ⁿ⁻² A and |adj(adj A)| = |A|^(n−1)²
adj(kA) = kⁿ⁻¹ adj A; adj(AB) = (adj B)(adj A); adj(Aᵀ) = (adj A)ᵀ
(AB)⁻¹ = B⁻¹A⁻¹; |A⁻¹| = 1/|A|

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation |A − λI| = 0. For a 2 × 2 matrix this is A² − (tr A)A + |A| I = O. Multiplying by A⁻¹ gives the inverse without computing cofactors.

5. System of Linear Equations

For three equations in x, y, z, write D (the coefficient determinant) and D_x, D_y, D_z (replace the corresponding column by the constants). By Cramer's rule, x = D_x/D, y = D_y/D, z = D_z/D.

Condition	Result
D ≠ 0	Unique solution (consistent)
D = 0 and at least one of D_x, D_y, D_z ≠ 0	No solution (inconsistent)
D = D_x = D_y = D_z = 0	Infinitely many solutions or no solution — check the equations directly
Homogeneous system (all constants 0)	Always has x = y = z = 0; non-trivial solutions exist only if D = 0

In matrix form AX = B: if |A| ≠ 0 then X = A⁻¹B.

6. Formula Master Table

Result (A is n × n)	Formula
Scalar multiple	\|kA\| = kⁿ\|A\|
Product	\|AB\| = \|A\|\|B\|
Inverse	A⁻¹ = adj A / \|A\|
Adjoint determinant	\|adj A\| = \|A\|ⁿ⁻¹
Double adjoint	adj(adj A) = \|A\|ⁿ⁻²A
Adjoint of kA	adj(kA) = kⁿ⁻¹ adj A
Transpose of product	(AB)ᵀ = BᵀAᵀ
Inverse of product	(AB)⁻¹ = B⁻¹A⁻¹
Cayley–Hamilton (2 × 2)	A² − (tr A)A + \|A\|I = O
Rotation matrix power	R(θ)ⁿ = R(nθ)
Symmetric matrices with entries from a k-element set (n × n)	k^n(n+1)/2
Skew-symmetric (odd order)	\|A\| = 0

7. Mind Map

Matrices & Determinants

Matrices

Types: symmetric, skew, orthogonal
AB ≠ BA
(AB)ᵀ = BᵀAᵀ

Determinants

Cofactor expansion
Row operations
|kA| = kⁿ|A|

Adjoint & Inverse

A adj A = |A| I
|adj A| = |A|ⁿ⁻¹
Cayley–Hamilton

Linear systems

Cramer's rule
D ≠ 0 unique
Homogeneous: D = 0 for non-trivial

8. Tips and Tricks

Shortcuts

For a rotation matrix R(θ) = [cos θ −sin θ; sin θ cos θ], R(θ)ⁿ = R(nθ). Reduce nθ modulo 2π first.
If AB = A and BA = B, then A and B are both idempotent (A² = A, B² = B).
Before expanding a 3 × 3 determinant, create zeros with row/column operations such as R₁ → R₁ + R₂ + R₃.
For consistency questions, first find the value of the parameter that makes D = 0, then compare two equations to see whether they coincide or contradict.
Counting questions (number of matrices with a property) are really permutation problems: count the free entries.

Mistakes students make

Writing |2A| = 2|A| for a 3 × 3 matrix. Correct: |2A| = 8|A|.
Using (A + B)² = A² + 2AB + B² — valid only when AB = BA.
Concluding "infinitely many solutions" just because D = D_x = D_y = D_z = 0 — the system may still be inconsistent.

9. Important One-liners

The diagonal entries of a skew-symmetric matrix are all zero.
The determinant of an orthogonal matrix is ±1.
If A is invertible, so are Aᵀ and adj A, with (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
A and Aᵀ have the same determinant and the same trace.
Only square matrices have determinants and (possibly) inverses.
If A² = I then A⁻¹ = A (involutory).
If |A| = 0 then A(adj A) = O.
For a 3 × 3 matrix, |adj A| = |A|² and |adj(adj A)| = |A|⁴.
The product of two symmetric matrices is symmetric only if they commute.

10. PYQ-pattern Questions with Solutions

These questions are original and modelled on concepts frequently tested in JEE Main.

Q1Numerical

A is a 3 × 3 matrix with |A| = 4. Find |adj(2A)|.

Show solution

adj(kA) = kⁿ⁻¹ adj A, so adj(2A) = 2² adj A = 4 adj A.

|adj(2A)| = |4 adj A| = 4³ |adj A| = 64 × |A|² = 64 × 16 = 1024.

Answer: 1024

Q2Medium

The system x + y + z = 6, x + 2y + 3z = 10, x + 2y + kz = μ has infinitely many solutions when:

k = 3, μ = 10
k = 3, μ ≠ 10
k ≠ 3
k = 2, μ = 10

Show solution

D = 1(2k − 6) − 1(k − 3) + 1(2 − 2) = k − 3. For infinitely many solutions we need D = 0, so k = 3.

With k = 3, the second and third equations have the same left side, x + 2y + 3z. They coincide only if μ = 10; then there are just two independent equations in three unknowns, giving infinitely many solutions. If μ ≠ 10 they contradict (no solution); if k ≠ 3 the solution is unique.

Answer: (A)

Q3Numerical

If A = [1 2; 3 4] and A² − 5A + kI = O, find k.

Show solution

By Cayley–Hamilton, A² − (tr A)A + |A|I = O. Here tr A = 5 and |A| = 4 − 6 = −2.

So A² − 5A − 2I = O, giving k = −2.

Answer: −2

Q4Medium

If A = [cos θ −sin θ; sin θ cos θ] with θ = π/4, then A²⁰²⁶ equals:

I
−I
[0 −1; 1 0]
[0 1; −1 0]

Show solution

A is a rotation matrix, so Aⁿ is rotation by nθ. 2026 × π/4 = 506.5π = 253 × 2π + π/2.

So A²⁰²⁶ = R(π/2) = [cos π/2 −sin π/2; sin π/2 cos π/2] = [0 −1; 1 0].

Answer: (C)

Q5Numerical

How many 3 × 3 symmetric matrices can be formed with every entry equal to 0 or 1?

Show solution

In a symmetric matrix the entries on and above the diagonal decide everything: 3 diagonal + 3 above-diagonal = 6 free entries, each with 2 choices.

Number = 2⁶ = 64.

Answer: 64

Q6Easy

The value of the determinant |1 1 1; a b c; a² b² c²| is:

(a − b)(b − c)(c − a)
(a + b)(b + c)(c + a)
abc
0

Show solution

Apply C₁ → C₁ − C₂ and C₂ → C₂ − C₃. The first row becomes (0, 0, 1), and we can take (a − b) and (b − c) common. The remaining 2 × 2 determinant is |1 1; a + b b + c| = (b + c) − (a + b) = c − a.

Result: (a − b)(b − c)(c − a) — the Vandermonde determinant.

Answer: (A)

Q7Hard

If A and B are square matrices of the same order such that AB = A and BA = B, then A² + B² equals:

A + B
2AB
AB + BA
O

Show solution

Using AB = A: A² = A·A = (AB)A = A(BA) = AB = A. Similarly, using BA = B: B² = (BA)B = B(AB) = BA = B.

So A² + B² = A + B.

Answer: (A)

Q8Hard

If A is a 3 × 3 skew-symmetric matrix and B = A + Aᵀ + 2I, then |A| + |B| equals:

Show solution

A is skew-symmetric of odd order, so |A| = 0.

Also Aᵀ = −A, so B = A − A + 2I = 2I, and |B| = |2I| = 2³ = 8.

|A| + |B| = 0 + 8 = 8.

Answer: (C) 8

11. Quick Revision Summary

Remember before the exam

|kA| = kⁿ|A|, |AB| = |A||B|, |adj A| = |A|ⁿ⁻¹, adj(kA) = kⁿ⁻¹adj A.
A⁻¹ = adj A/|A| exists only if |A| ≠ 0; (AB)⁻¹ = B⁻¹A⁻¹.
Cayley–Hamilton: A² − (tr A)A + |A|I = O for 2 × 2 matrices.
D ≠ 0 → unique solution; D = 0 → none or infinitely many; homogeneous non-trivial needs D = 0.
Skew-symmetric of odd order: |A| = 0; orthogonal: |A| = ±1; rotation: R(θ)ⁿ = R(nθ).

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