Maths · JEE Main & Advanced 2026 · Matrices

Matrices notes, formulas & PYQs for JEE 2026

Complete chapter — types of matrices, algebra, transpose, trace, symmetric & skew-symmetric matrices, special matrices (idempotent, nilpotent, orthogonal), adjoint, inverse, elementary operations, system of linear equations, 100+ worked examples, and JEE PYQs with solutions.

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Matrices JEE Notes, Formulas, PYQs
Matrices — JEE Notes, Formulas, PYQs
1

Matrix Basics and Types

A Matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices provide a compact way to represent systems of linear equations and transformations. In JEE, this chapter typically contributes 2-3 questions, making it highly scoring if concepts are clear.

1.1 Definition and Order

Matrix Notation

A matrix $A$ of order $m \times n$ is represented as:

\[A = [a_{ij}]_{m \times n}\]

where $1 \le i \le m$ (row index) and $1 \le j \le n$ (column index). The element $a_{ij}$ belongs to the $i$-th row and $j$-th column.

1.2 Types of Matrices

TypeConditionExample / Property
Row Matrix$m = 1$Order $1 \times n$
Column Matrix$n = 1$Order $m \times 1$
Zero / Null Matrix (O)All $a_{ij} = 0$Denoted by $O$
Square Matrix$m = n$Order $n \times n$
Diagonal MatrixSquare matrix with $a_{ij} = 0$ for $i \neq j$Only principal diagonal has non-zero entries
Scalar MatrixDiagonal matrix where $a_{ii} = k$ (constant)$a_{ij} = 0$ ($i \neq j$), $a_{ii} = k$
Identity Matrix (I)Scalar matrix with $k = 1$$a_{ii} = 1$, $a_{ij} = 0$ ($i \neq j$)
Upper Triangular$a_{ij} = 0$ for $i > j$Entries below diagonal are zero
Lower Triangular$a_{ij} = 0$ for $i < j$Entries above diagonal are zero

Important Note

  • Only square matrices have determinants, inverses, and traces.
  • The diagonal elements of a square matrix $A=[a_{ij}]_{n \times n}$ are $a_{11}, a_{22}, \ldots, a_{nn}$.
  • Identity matrix is always denoted by $I_n$ for order $n$.
2

Algebra of Matrices

Matrices can be added, subtracted, and multiplied under specific conditions. Matrix algebra is a core area in JEE, with questions frequently testing properties of matrix multiplication.

2.1 Addition and Scalar Multiplication

Addition: Two matrices $A$ and $B$ can be added only if they have the same order. $A+B = [a_{ij} + b_{ij}]$.

Scalar Multiplication: $kA = [k \cdot a_{ij}]$ (multiply every element by $k$).

2.2 Matrix Multiplication

Most Repeated PYQ — Multiplication Condition

Two matrices $A$ of order $m \times n$ and $B$ of order $n \times p$ can be multiplied to give a matrix $C$ of order $m \times p$.

\[C = AB \implies c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}\]

Condition: Number of columns in $A$ = Number of rows in $B$.

Critical Properties

  • Not Commutative: Generally $AB \neq BA$.
  • Associative: $(AB)C = A(BC)$.
  • Distributive: $A(B+C) = AB + AC$.
  • $AB = 0$ does not imply $A=0$ or $B=0$.
  • $AB = AC$ does not imply $B=C$ (Cancellation law fails).

PYQ — JEE Main 2023 (Apr 10 Shift 1)

Question: If $A$ and $B$ are two $3 \times 3$ matrices such that $AB = A$ and $BA = B$, then $A^2 + B^2$ is equal to:

(a) $2A$

(b) $2B$

(c) $A + B$

(d) $AB$

Solution:

Given: $AB = A$ and $BA = B$.

Multiply $AB = A$ by $A$ on the right:

\[(AB)A = A \cdot A \implies A(BA) = A^2\]

Since $BA = B$, substitute:

\[A(B) = A^2 \implies AB = A^2\]

But $AB = A$, so $A^2 = A$.

Similarly, multiply $BA = B$ by $B$ on the right:

\[(BA)B = B \cdot B \implies B(AB) = B^2\]

Since $AB = A$, substitute:

\[B(A) = B^2 \implies BA = B^2\]

But $BA = B$, so $B^2 = B$.

Therefore:

\[A^2 + B^2 = A + B\]

Answer: (c) $A + B$

3

Trace and Transpose of a Matrix

The trace is a scalar property of square matrices, while the transpose rearranges the elements. Both have properties that frequently appear in JEE Main objective questions.

3.1 Trace of a Matrix

Important — Trace Properties

The trace of a square matrix $A = [a_{ij}]_{n \times n}$ is the sum of its principal diagonal elements:

\[\text{tr}(A) = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}\]
  • $\text{tr}(A+B) = \text{tr}(A) + \text{tr}(B)$
  • $\text{tr}(kA) = k \cdot \text{tr}(A)$
  • $\text{tr}(AB) = \text{tr}(BA)$ (Crucial Property!)
  • $\text{tr}(I_n) = n$

3.2 Transpose of a Matrix

The transpose of matrix $A$, denoted $A'$ or $A^T$, is obtained by interchanging rows and columns:

\[(A^T)_{ij} = a_{ji}\]

If $A$ is $m \times n$, then $A^T$ is $n \times m$.

3.3 Properties of Transpose

  • $(A^T)^T = A$
  • $(A+B)^T = A^T + B^T$
  • $(kA)^T = k A^T$
  • $(AB)^T = B^T A^T$ (Reversal Law)

PYQ — JEE Main 2024 (Jan 31 Shift 1)

Question: Let $A$ be a $2 \times 2$ matrix such that $A^2 = I$. If the trace of $A$ is $-2$, then $A$ is equal to:

(a) $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$

(b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

(c) $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$

(d) $\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}$

Solution:

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

Given trace $a + d = -2$, and $A^2 = I$.

By Cayley-Hamilton Theorem (or direct expansion): $A^2 - (a+d)A + (ad-bc)I = 0$.

Given $A^2 = I \implies I - (-2)A + (ad-bc)I = 0$.

\[I + 2A + (ad-bc)I = 0\]
\[2A = -(1 + ad - bc)I\]
\[A = -\frac{1 + \det(A)}{2} I\]

Since $A$ is a scalar multiple of $I$, it must be diagonal. Let $A = \begin{bmatrix} \lambda & 0 \\ 0 & \mu \end{bmatrix}$.

Then $A^2 = \begin{bmatrix} \lambda^2 & 0 \\ 0 & \mu^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \implies \lambda = \pm 1, \mu = \pm 1$.

But trace $= \lambda + \mu = -2$. This is only possible if $\lambda = -1$ and $\mu = -1$.

\[A = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\]

Answer: (a)

4

Symmetric and Skew-Symmetric Matrices

Special matrix types based on transpose are heavily tested in JEE. The decomposition theorem is a favorite concept for examiners.

4.1 Definitions

Most Repeated PYQ — Symmetric & Skew-Symmetric

Symmetric Matrix: A square matrix $A$ is symmetric if $A^T = A$.

  • Condition: $a_{ij} = a_{ji}$ for all $i, j$.

Skew-Symmetric Matrix: A square matrix $A$ is skew-symmetric if $A^T = -A$.

  • Condition: $a_{ij} = -a_{ji}$ for all $i, j$.
  • Important: Diagonal elements must be zero ($a_{ii} = -a_{ii} \implies a_{ii} = 0$).

4.2 The Decomposition Theorem

Important — Matrix Decomposition

Every square matrix $A$ can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix:

\[A = P + Q\]

where $P$ is symmetric and $Q$ is skew-symmetric:

\[P = \frac{1}{2}(A + A^T), \quad Q = \frac{1}{2}(A - A^T)\]

4.3 Properties

  • If $A$ and $B$ are symmetric, $A \pm B$ and $kA$ are symmetric.
  • $AB$ is symmetric iff $A$ and $B$ commute ($AB = BA$).
  • For any square matrix $A$, $A + A^T$ is symmetric and $A - A^T$ is skew-symmetric.
  • $A^T A$ and $A A^T$ are always symmetric for any matrix $A$.

PYQ — JEE Main 2023 (Apr 13 Shift 2)

Question: Let $A$ be a $3 \times 3$ symmetric matrix. If $|A| = 2$, then find $|A^{-1} A^T|$.

(a) 2

(b) 1/2

(c) 1

(d) 4

Solution:

Given $A$ is symmetric, so $A^T = A$.

Substitute $A^T = A$ into the expression:

\[|A^{-1} A^T| = |A^{-1} A|\]
\[= |I| = 1\]
\[|A^{-1} A^T| = 1\]

Answer: (c) 1

5

Special Matrices

JEE frequently tests your knowledge of matrices defined by specific algebraic properties.

5.1 Idempotent, Nilpotent, and Involutary Matrices

Important — Special Matrix Definitions

  • Idempotent Matrix: $A^2 = A$
  • Nilpotent Matrix: $A^k = 0$ for some positive integer $k$. The smallest such $k$ is the index of nilpotency.
  • Involutary Matrix: $A^2 = I \implies A = A^{-1}$

5.2 Orthogonal Matrix

Important — Orthogonal Matrix

A square matrix $A$ is orthogonal if:

\[A A^T = A^T A = I\]

This implies $A^T = A^{-1}$.

Determinant: If $A$ is orthogonal, $|A| = \pm 1$.

PYQ — JEE Main 2019 (Apr 8 Shift 1)

Question: If $A$ is a $3 \times 3$ matrix such that $A^2 = A$ and $|A| \neq 0$, then find $|2A - I|$.

(a) 1

(b) -1

(c) 8

(d) 0

Solution:

Given $A^2 = A$.

Since $|A| \neq 0$, $A$ is invertible. Multiply both sides by $A^{-1}$:

\[A^2 A^{-1} = A A^{-1}\]
\[A = I\]

So $A$ is the identity matrix.

Now substitute $A = I$ into $|2A - I|$:

\[|2I - I| = |I| = 1\]
\[|2A - I| = 1\]

Answer: (a) 1

6

Adjoint and Inverse of a Matrix

The adjoint and inverse are foundational for solving matrix equations and systems of linear equations. The properties of adjoints are among the most frequently tested concepts in JEE Main.

6.1 Adjoint of a Matrix

The adjoint of a square matrix $A$, denoted $\text{adj}(A)$, is the transpose of the cofactor matrix $C$.

\[\text{adj}(A) = C^T\]

Most Repeated PYQ — Adjoint Properties

  • $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$
  • $|\text{adj}(A)| = |A|^{n-1}$ where $n$ is the order of matrix $A$.
  • $\text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A)$
  • $\text{adj}(kA) = k^{n-1} \text{adj}(A)$
  • $\text{adj}(\text{adj}(A)) = |A|^{n-2} A$ (for $n \ge 3$)

6.2 Inverse of a Matrix

Important — Inverse Formula

A square matrix $A$ is invertible iff $|A| \neq 0$. The inverse $A^{-1}$ is given by:

\[A^{-1} = \frac{\text{adj}(A)}{|A|}\]

6.3 Properties of Inverse

  • $(A^{-1})^{-1} = A$
  • $(AB)^{-1} = B^{-1} A^{-1}$ (Reversal Law)
  • $(A^T)^{-1} = (A^{-1})^T$
  • $|A^{-1}| = \frac{1}{|A|}$

PYQ — JEE Main 2023 (Apr 11 Shift 2)

Question: Let $A$ be a $3 \times 3$ matrix with $|A| = 2$. Find $|2 \cdot \text{adj}(A)|$.

(a) 16

(b) 32

(c) 8

(d) 4

Solution:

We know the property: $|k \cdot A| = k^n |A|$.

Applying this to the adjoint matrix (which is $3 \times 3$):

\[|2 \cdot \text{adj}(A)| = 2^3 \cdot |\text{adj}(A)|\]
\[= 8 \cdot |\text{adj}(A)|\]

Using property $|\text{adj}(A)| = |A|^{n-1}$, where $n=3$ and $|A|=2$:

\[|\text{adj}(A)| = 2^{3-1} = 2^2 = 4\]

Therefore:

\[|2 \cdot \text{adj}(A)| = 8 \times 4 = 32\]
\[|2 \cdot \text{adj}(A)| = 32\]

Answer: (b) 32

7

Elementary Operations & System of Equations

Elementary operations provide a systematic way to find inverses and solve systems of equations. The matrix method for solving linear equations is a highly tested application.

7.1 Elementary Operations

Any matrix can be transformed using elementary row/column operations:

  • Interchange: $R_i \leftrightarrow R_j$
  • Multiplication: $R_i \to k R_i$ ($k \neq 0$)
  • Addition: $R_i \to R_i + k R_j$

7.2 Finding Inverse Using Elementary Operations

Procedure

To find $A^{-1}$, write the augmented matrix $[A | I]$ and apply row operations until $A$ becomes $I$. The right side automatically becomes $A^{-1}$:

\[[A | I] \xrightarrow{\text{row ops}} [I | A^{-1}]\]

7.3 System of Linear Equations (AX = B)

Most Repeated PYQ — Consistency Conditions

For a system $AX = B$ where $A$ is an $n \times n$ matrix:

  • Unique Solution: $|A| \neq 0$ (Consistent). Solution is $X = A^{-1}B$.
  • No Solution: $|A| = 0$ and $(\text{adj} A)B \neq 0$ (Inconsistent).
  • Infinite Solutions: $|A| = 0$ and $(\text{adj} A)B = 0$ (Consistent).

PYQ — JEE Main 2024 (Jan 29 Shift 2)

Question: If the system of equations $x + y + z = 6$, $x + 2y + 3z = 10$, $x + 2y + \lambda z = \mu$ has infinitely many solutions, then $\lambda + \mu$ is equal to:

(a) 10

(b) 13

(c) 11

(d) 12

Solution:

For infinite solutions, the third equation must be a linear combination of the first two, meaning $|A| = 0$ and rank remains 2.

Subtract Eq 2 from Eq 3:

\[(x+2y+\lambda z) - (x+2y+3z) = \mu - 10\]
\[(\lambda - 3)z = \mu - 10\]

For this to hold for infinite values of $z$ (infinite solutions), both coefficients must be zero:

\[\lambda - 3 = 0 \implies \lambda = 3\]
\[\mu - 10 = 0 \implies \mu = 10\]

Therefore:

\[\lambda + \mu = 3 + 10 = 13\]

Answer: (b) 13

Common Mistakes Summary

  • Mistake: Assuming $AB=0 \implies A=0$ or $B=0$ (Matrix multiplication lacks zero divisors property).
  • Mistake: Forgetting that diagonal elements of a skew-symmetric matrix are strictly zero.
  • Mistake: Using $|kA| = k|A|$ instead of $|kA| = k^n|A|$ for an $n \times n$ matrix.
  • Mistake: Confusing $|\text{adj}(A)| = |A|^{n-1}$ with $|\text{adj}(A)| = |A|^{n-2}$.

Previous Year Questions Analysis

JEE Main (Last 10 Years)

  • ✓ Adjoint and Inverse Properties: 22%
  • ✓ Matrix Multiplication Properties: 18%
  • ✓ System of Equations Consistency: 16%
  • ✓ Symmetric & Skew-Symmetric: 15%
  • ✓ Trace and Transpose: 12%
  • ✓ Special Matrices (Idempotent, etc.): 10%
  • ✓ Elementary Operations: 7%

JEE Advanced (Last 10 Years)

  • ✓ Multi-concept Matrix Algebra: 25%
  • ✓ System of Equations (Parameter based): 20%
  • ✓ Adjoint Properties (n=3,4): 18%
  • ✓ Idempotent & Nilpotent Proofs: 15%
  • ✓ Orthogonal Matrices: 12%
  • ✓ Finding Inverse (Elementary): 10%

Top 12 Most Repeated Question Types

  1. Finding $|k \cdot \text{adj}(A)|$ or $|\text{adj}(\text{adj}(A))|$ given $|A|$.
  2. Solving $AX=B$ and checking for infinite/no solutions based on parameters.
  3. Expressing a matrix as sum of symmetric and skew-symmetric matrices.
  4. Finding inverse of a $2 \times 2$ or $3 \times 3$ matrix using elementary operations.
  5. Using property $\text{tr}(AB) = \text{tr}(BA)$ to find unknowns.
  6. Verifying if $AB=AC \implies B=C$.
  7. Finding values of $x, y, z$ if a matrix is symmetric or skew-symmetric.
  8. Properties of Idempotent ($A^2=A$) and Involutary ($A^2=I$) matrices.
  9. Proving $(AB)^T = B^T A^T$ for specific matrices.
  10. Finding the matrix polynomial (e.g., $A^2 - 5A + 7I = 0$).
  11. Determinant of an orthogonal matrix.
  12. Trace of a special matrix.

Weightage Analysis

JEE Main: 8-12 marks (2-3 questions)
JEE Advanced: 4-8 marks (1-2 questions)
Difficulty Level: Easy to Moderate
Time Required: 3-4 hours practice
Download Complete PYQ PDF (2015-2024)

Practice Problem Set

Level 1: Basic (JEE Main Standard)

  1. Find the values of $x, y, z$ if $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ is skew-symmetric.
  2. If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A^{-1}$ using the adjoint method.
  3. If $A$ is a $3 \times 3$ matrix and $|A| = 5$, find $|\text{adj}(\text{adj} A)|$.
  4. Find the trace of $A^2$ if $A = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}$.
  5. Express $A = \begin{bmatrix} 2 & -4 \\ 3 & 5 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
  6. Solve the system: $x+y+z=3, 2x+3y+4z=10, x+2y+3z=7$.

Level 2: Intermediate (JEE Main/Advanced)

  1. If $A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, find $A^n$.
  2. Find the inverse of $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$ using elementary row operations.
  3. If $A^2 = A$ and $|A| \neq 0$, find $A$.
  4. Let $A$ be a $3\times 3$ orthogonal matrix with $|A| = -1$. Find $|A - I|$.
  5. If $A$ and $B$ are square matrices of the same order, prove that $\text{tr}(AB) = \text{tr}(BA)$.
  6. Find the values of $a$ and $b$ for which the system $x+y+z=1, 2x+3y+4z=a, x+ay+3z=b$ has infinite solutions.
Get Solutions PDF with Step-by-Step Working

> FAQ

Frequently Asked Questions

What is the formula for the inverse of a matrix?

The inverse of a square matrix A is given by A⁻¹ = adj(A) / |A|, where adj(A) is the adjoint (transpose of cofactor matrix) and |A| is the determinant of A. The inverse exists if and only if |A| ≠ 0 (i.e., A is non-singular).

What is the difference between symmetric and skew-symmetric matrices?

A square matrix A is symmetric if A' = A (i.e., a_ij = a_ji). It is skew-symmetric if A' = -A (i.e., a_ij = -a_ji). For a skew-symmetric matrix, all principal diagonal elements must be zero. Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

How do you find the inverse of a matrix using elementary operations?

To find A⁻¹ using elementary row operations, write the augmented matrix [A | I] and apply row operations to convert the left side A into the identity matrix I. The right side automatically transforms into A⁻¹. Alternatively, you can use column operations by writing A and I as [A / I].

What are the properties of adjoint of a matrix?

Key properties of adjoint for an n×n matrix A: 1) A(adj A) = (adj A)A = |A|I, 2) |adj A| = |A|^(n-1), 3) adj(AB) = adj(B)adj(A), 4) adj(kA) = k^(n-1) adj(A), 5) adj(adj A) = |A|^(n-2) A.

What is the condition for a system of equations to have infinite solutions?

For a non-homogeneous system of linear equations AX = B, if |A| = 0 and (adj A)B = 0, the system has infinitely many solutions. If |A| = 0 and (adj A)B ≠ 0, it has no solution. If |A| ≠ 0, it has a unique solution given by X = A⁻¹B.

What is a nilpotent matrix?

A square matrix A is called nilpotent if there exists a positive integer k such that Aᵏ = 0. The smallest such k is called the index of nilpotency. The determinant and trace of a nilpotent matrix are always zero.

What is the trace of a matrix?

The trace of a square matrix A is the sum of its principal diagonal elements. Key properties include tr(A+B) = tr(A) + tr(B) and tr(AB) = tr(BA).

How many questions come from Matrices in JEE Mains?

On average 2-3 questions (8-12 marks) appear from Matrices in every JEE Main session. The most repeated topics are finding adjoint/inverse using properties, symmetric/skew-symmetric decomposition, matrix multiplication properties, and solving systems of equations.

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Complete Matrices Mastery Guide for JEE 2026

Why This Chapter is Important

Matrices is one of the most scoring chapters in JEE Mathematics:

  • 2-3 questions in almost every JEE Main session (8-12 marks)
  • Largely formula-based — quick to solve if properties are memorized
  • Connects with Determinants and System of Equations
  • Adjoint properties like $|\text{adj} A| = |A|^{n-1}$ appear almost every year
  • System of equations matrix method is highly scoring

Topic-wise Weightage Analysis

Adjoint & Inverse Properties 22%
Algebra & Multiplication 18%
System of Equations 16%
Symmetric/Skew-Symmetric 15%
Other (Trace, Special) 29%

Must-Remember Formulas & Values

Trace & Transpose

tr(A+B) = tr(A) + tr(B)

tr(AB) = tr(BA)

(AB)' = B'A'

Matrix Decomposition

A = P + Q

P = ½(A + A') [Symmetric]

Q = ½(A - A') [Skew-Symmetric]

Adjoint & Inverse

A(adj A) = |A|I

A⁻¹ = adj(A) / |A|

|adj A| = |A|ⁿ⁻¹

adj(adj A) = |A|ⁿ⁻² A

|kA| = kⁿ |A|

System AX = B

|A| ≠ 0 ⟹ Unique Solution

|A| = 0 & (adj A)B = 0 ⟹ Infinite

|A| = 0 & (adj A)B ≠ 0 ⟹ No Solution

5-Step Exam Strategy

  1. 1
    For adjoint problems: Always remember |adj A| = |A|^(n-1) and adj(kA) = k^(n-1) adj A.
  2. 2
    For symmetric/skew-symmetric: Use A = ½(A+A') + ½(A-A'). Diagonals of skew-symmetric are always 0.
  3. 3
    For system of equations: Check |A| first. If 0, check (adj A)B.
  4. 4
    For special matrices: Orthogonal ⟹ |A|=±1, Idempotent ⟹ A²=A, Involutary ⟹ A²=I.
  5. 5
    For finding inverse: Use [A|I] row operations if standard adjoint method is too slow for 3x3.

7-Day Mastery Roadmap

Day 1

Basics & Types

  • • Order & Types
  • • Equality
  • • Scalar Mult
Day 2

Algebra

  • • Matrix Mult
  • • Properties
  • • Practice 30 Qs
Day 3

Transpose

  • • Trace prop.
  • • Transpose
  • • (AB)' = B'A'
Day 4

Sym/Skew

  • • Definitions
  • • Decomposition
  • • Properties
Day 5

Special

  • • Idempotent
  • • Orthogonal
  • • Nilpotent
Day 6

Adjoint

  • • adj A formulas
  • • |adj A|
  • • Inverse
Day 7

Systems

  • • AX=B
  • • Consistency
  • • PYQs

Matrices — Complete JEE Study Guide

Matrices is a foundational chapter in JEE Mathematics, introducing students to rectangular arrays of numbers and their algebraic properties. The chapter is largely formula-driven and accounts for 2-3 questions (8-12 marks) in nearly every JEE Main session.

The most important topics for JEE from this chapter are adjoint and inverse properties (formulas like $|\text{adj}(A)| = |A|^{n-1}$ and $\text{adj}(\text{adj}(A)) = |A|^{n-2}A$ are tested almost every year), symmetric and skew-symmetric matrices (the decomposition theorem $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$ is crucial), and system of linear equations (consistency conditions using matrix method).

Students should focus on mastering matrix multiplication properties, particularly the fact that matrix multiplication is not commutative ($AB \neq BA$) and does not obey the zero-product property ($AB=0$ does not imply $A=0$ or $B=0$). The trace property $\text{tr}(AB) = \text{tr}(BA)$ is another frequently tested concept. For special matrices, understanding the conditions for Idempotent ($A^2=A$), Nilpotent ($A^k=0$), and Orthogonal ($AA^T=I$) matrices is essential.

When solving systems of equations $AX = B$, always check the determinant first. If $|A| \neq 0$, the system has a unique solution $X = A^{-1}B$. If $|A| = 0$, check $(\text{adj} A)B$: if it is zero, there are infinite solutions; if non-zero, no solution exists. Finding the inverse using elementary row operations $[A|I] \to [I|A^{-1}]$ is a standard 5-mark question type that requires speed and accuracy.