The Hook: The Matrix in Your Pocket
Ever wondered how Instagram applies filters to your photos in milliseconds? When you select “Nashville” or “Clarendon,” your phone multiplies a 3×3 color transformation matrix with millions of pixels simultaneously!
Each pixel’s RGB values are arranged in a matrix, and a transformation matrix changes the colors:
$$\begin{bmatrix} R' \\ G' \\ B' \end{bmatrix} = \begin{bmatrix} 1.2 & 0.1 & 0 \\ 0 & 1.1 & 0.2 \\ 0.1 & 0 & 0.9 \end{bmatrix} \begin{bmatrix} R \\ G \\ B \end{bmatrix}$$Real applications: Computer graphics, 3D game rotations, Google’s PageRank algorithm, AI neural networks, encryption, and even solving traffic flow problems — all use matrices!
Why this matters for JEE: Matrices appear in 3-4 questions in JEE Main and form the foundation for determinants, linear equations, and advanced coordinate geometry in JEE Advanced.
Prerequisites
Before diving into matrices, you should be comfortable with:
- Sets and Relations — Understanding of ordered pairs
- Basic algebra — Working with variables and equations
- Array/table representation — Organizing data in rows and columns
Interactive Demo: Matrix Visualization
Create Your Own Matrix
Click on elements to change values and see how matrix notation works!
What is a Matrix?
Definition
A matrix is a rectangular array of numbers (or expressions) arranged in rows and columns, enclosed in brackets.
$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}$$Components:
- Elements/Entries: The individual numbers $a_{ij}$ where $i$ = row number, $j$ = column number
- Rows: Horizontal lines of elements
- Columns: Vertical lines of elements
Order of a Matrix
The order (or dimension) of a matrix is given by:
$$\boxed{\text{Order} = m \times n}$$where:
- $m$ = number of rows
- $n$ = number of columns
Read as: “m by n” or “m cross n”
Examples
Order: $2 \times 3$ (2 rows, 3 columns)
Elements: $a_{11} = 1, a_{12} = 2, a_{13} = 3, a_{21} = 4, a_{22} = 5, a_{23} = 6$
Order: $3 \times 1$ (3 rows, 1 column)
Notation and Element Reference
Standard Notation
A matrix is denoted by:
- Capital letters: $A, B, C, M, X$
- Elements denoted by corresponding lowercase letters with subscripts: $a_{ij}, b_{ij}$
Subscript Convention
For element $a_{ij}$:
- First subscript $i$ → row number
- Second subscript $j$ → column number
Think “RC” = Row first, Column second
Just like drinking RC Cola: R (row) comes before C (column)!
$a_{23}$ means: Row 2, Column 3 (not the other way around)
Compact Notation
A matrix $A$ with elements $a_{ij}$ can be written as:
$$A = [a_{ij}]_{m \times n}$$Example: $A = [2i + 3j]_{2 \times 3}$ means:
$$A = \begin{bmatrix} 2(1) + 3(1) & 2(1) + 3(2) & 2(1) + 3(3) \\ 2(2) + 3(1) & 2(2) + 3(2) & 2(2) + 3(3) \end{bmatrix} = \begin{bmatrix} 5 & 8 & 11 \\ 7 & 10 & 13 \end{bmatrix}$$Types of Matrices
1. Row Matrix
A matrix with only one row.
Order: $1 \times n$
Example: $A = \begin{bmatrix} 1 & -2 & 5 & 7 \end{bmatrix}$ is a $1 \times 4$ row matrix.
2. Column Matrix
A matrix with only one column.
Order: $m \times 1$
Example: $B = \begin{bmatrix} 3 \\ -1 \\ 0 \end{bmatrix}$ is a $3 \times 1$ column matrix.
3. Square Matrix
A matrix where number of rows = number of columns.
Order: $n \times n$
Example: $C = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is a $2 \times 2$ square matrix.
Important terms for square matrices:
Principal/Main Diagonal: Elements $a_{ii}$ where row number = column number
- For $2 \times 2$: Elements $a_{11}, a_{22}$
- For $3 \times 3$: Elements $a_{11}, a_{22}, a_{33}$
Trace: Sum of diagonal elements
$$\boxed{\text{tr}(A) = \sum_{i=1}^{n} a_{ii}}$$Example: For $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, $\text{tr}(A) = 1 + 5 + 9 = 15$
4. Diagonal Matrix
A square matrix where all non-diagonal elements are zero.
$$a_{ij} = 0 \text{ when } i \neq j$$Example: $D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 2 \end{bmatrix}$
Compact notation: $D = \text{diag}(3, -5, 2)$
Diagonal elements can be zero! As long as all off-diagonal elements are zero, it’s a diagonal matrix.
$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ is still diagonal.
5. Scalar Matrix
A diagonal matrix where all diagonal elements are equal.
$$a_{ij} = \begin{cases} k & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$Example: $S = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix} = 5I$
6. Identity Matrix (Unit Matrix)
A scalar matrix where all diagonal elements equal 1.
Denoted by $I$ or $I_n$ (for $n \times n$ identity matrix)
$$I_n = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$$Examples:
$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$Property: $AI = IA = A$ (Identity is the “1” of matrix multiplication)
7. Zero/Null Matrix
A matrix where all elements are zero.
Denoted by $O$ or $0$
Example: $O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is a $2 \times 3$ null matrix.
Property: $A + O = O + A = A$ (Zero matrix is the additive identity)
8. Upper Triangular Matrix
A square matrix where all elements below the main diagonal are zero.
$$a_{ij} = 0 \text{ when } i > j$$Example: $U = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$
9. Lower Triangular Matrix
A square matrix where all elements above the main diagonal are zero.
$$a_{ij} = 0 \text{ when } i < j$$Example: $L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}$
10. Symmetric Matrix
A square matrix where $A = A^T$ (matrix equals its transpose).
$$a_{ij} = a_{ji} \text{ for all } i, j$$Example: $S = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$
Notice: Elements are symmetric about the main diagonal.
Visual check: Mirror image across diagonal should be identical.
11. Skew-Symmetric Matrix
A square matrix where $A = -A^T$ (matrix equals negative of its transpose).
$$a_{ij} = -a_{ji} \text{ for all } i, j$$Important consequence: All diagonal elements must be zero! (Since $a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0$)
Example: $K = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$
Symmetric: Like looking in a mirror — same reflection
- $A = A^T$
- Elements: $a_{ij} = a_{ji}$
- Diagonal can be anything
Skew-Symmetric: Like looking in an evil mirror — opposite reflection
- $A = -A^T$
- Elements: $a_{ij} = -a_{ji}$
- Diagonal MUST be zero
Quick test: Check element $(1,2)$ and $(2,1)$:
- If they’re equal → might be symmetric
- If they’re negatives → might be skew-symmetric
Equality of Matrices
Two matrices $A$ and $B$ are equal if and only if:
- They have the same order ($m \times n$)
- All corresponding elements are equal: $a_{ij} = b_{ij}$ for all $i, j$
Example
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 2 & 0 \\ 3 & 4 & 0 \end{bmatrix}$$- $A = B$ ✓ (same order, same elements)
- $A \neq C$ ✗ (different order)
Left is $1 \times 3$ (row matrix), right is $3 \times 1$ (column matrix).
Even though they have the same numbers, they are NOT equal because orders differ!
Properties Summary
| Matrix Type | Key Property | Diagonal Elements | Example Order |
|---|---|---|---|
| Row | 1 row | N/A | $1 \times n$ |
| Column | 1 column | N/A | $m \times 1$ |
| Square | Rows = Columns | Any values | $n \times n$ |
| Diagonal | Off-diagonal = 0 | Any values | $n \times n$ |
| Scalar | Diagonal all equal | All equal | $n \times n$ |
| Identity | Diagonal all 1, rest 0 | All = 1 | $n \times n$ |
| Null | All elements = 0 | All = 0 | $m \times n$ |
| Upper Triangular | Below diagonal = 0 | Any values | $n \times n$ |
| Lower Triangular | Above diagonal = 0 | Any values | $n \times n$ |
| Symmetric | $A = A^T$ | Any values | $n \times n$ |
| Skew-Symmetric | $A = -A^T$ | All = 0 | $n \times n$ |
Common Mistakes to Avoid
Wrong: $A$ is $3 \times 2$ means 3 columns and 2 rows
Right: $A$ is $3 \times 2$ means 3 rows and 2 columns
Remember: “Rows × Columns” (RC Cola!)
Wrong: All diagonal elements must be non-zero in a diagonal matrix
Right: Diagonal elements can be zero! The requirement is only that off-diagonal elements are zero.
$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ is a valid diagonal matrix.
Wrong: Diagonal elements must be zero in symmetric matrices
Right: Only skew-symmetric matrices require zero diagonal. Symmetric matrices can have any diagonal values!
Symmetric: $\begin{bmatrix} 5 & 2 \\ 2 & 3 \end{bmatrix}$ ✓
Skew-symmetric: $\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$ ✓ (must have zero diagonal)
Wrong: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $a_{21} = 2$
Right: $a_{21} = 3$ (row 2, column 1)
First subscript is always row, second is always column!
Quick Identification Flowchart
graph TD
A[Matrix] --> B{Square?}
B -->|No| C{1 row?}
B -->|Yes| D{All zeros?}
C -->|Yes| E[Row Matrix]
C -->|No| F{1 column?}
F -->|Yes| G[Column Matrix]
F -->|No| H[Rectangular Matrix]
D -->|Yes| I[Null Matrix]
D -->|No| J{Off-diagonal zeros?}
J -->|Yes| K{Diagonal all equal?}
J -->|No| L{A = A^T?}
K -->|Yes| M{All diagonal = 1?}
K -->|No| N[Diagonal Matrix]
M -->|Yes| O[Identity Matrix]
M -->|No| P[Scalar Matrix]
L -->|Yes| Q[Symmetric]
L -->|No| R{A = -A^T?}
R -->|Yes| S[Skew-Symmetric]
R -->|No| T[General Square]Practice Problems
Level 1: Foundation (NCERT)
What is the order of matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{bmatrix}$?
Solution:
Count rows: 4 Count columns: 3
Answer: $4 \times 3$
For $A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$, find $a_{23}$ and $a_{32}$.
Solution:
$a_{23}$ = element in row 2, column 3 = 7
$a_{32}$ = element in row 3, column 2 = 9
Answer: $a_{23} = 7$, $a_{32} = 9$
Construct a $2 \times 3$ matrix $A = [a_{ij}]$ where $a_{ij} = i + j$.
Solution:
$a_{11} = 1 + 1 = 2$, $a_{12} = 1 + 2 = 3$, $a_{13} = 1 + 3 = 4$
$a_{21} = 2 + 1 = 3$, $a_{22} = 2 + 2 = 4$, $a_{23} = 2 + 3 = 5$
Answer: $A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix}$
Level 2: JEE Main
If $A = \begin{bmatrix} x+2 & 3 \\ 4 & y-1 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 3 \\ 4 & 2 \end{bmatrix}$, find $x$ and $y$ such that $A = B$.
Solution:
For $A = B$:
- $x + 2 = 5 \Rightarrow x = 3$
- $y - 1 = 2 \Rightarrow y = 3$
Answer: $x = 3$, $y = 3$
Identify the type of matrix: $A = \begin{bmatrix} 0 & 5 & -3 \\ -5 & 0 & 2 \\ 3 & -2 & 0 \end{bmatrix}$
Solution:
Check if symmetric: $a_{12} = 5$ but $a_{21} = -5$, so $a_{12} \neq a_{21}$. Not symmetric.
Check if skew-symmetric:
- $a_{12} = 5 = -a_{21} = -(-5)$ ✓
- $a_{13} = -3 = -a_{31} = -(3)$ ✓
- $a_{23} = 2 = -a_{32} = -(-2)$ ✓
- All diagonal elements are 0 ✓
Answer: Skew-symmetric matrix
Find the trace of $A = \begin{bmatrix} 2 & 3 & 1 \\ 0 & -1 & 4 \\ 5 & 2 & 3 \end{bmatrix}$.
Solution:
$\text{tr}(A) = a_{11} + a_{22} + a_{33} = 2 + (-1) + 3 = 4$
Answer: 4
Level 3: JEE Advanced
If $A$ is a $3 \times 3$ matrix such that $A = [a_{ij}]$ where $a_{ij} = i^2 - j^2$, determine whether $A$ is symmetric, skew-symmetric, or neither.
Solution:
First, construct the matrix:
- $a_{11} = 1^2 - 1^2 = 0$, $a_{12} = 1^2 - 2^2 = -3$, $a_{13} = 1^2 - 3^2 = -8$
- $a_{21} = 2^2 - 1^2 = 3$, $a_{22} = 2^2 - 2^2 = 0$, $a_{23} = 2^2 - 3^2 = -5$
- $a_{31} = 3^2 - 1^2 = 8$, $a_{32} = 3^2 - 2^2 = 5$, $a_{33} = 3^2 - 3^2 = 0$
Check: $a_{12} = -3$ and $a_{21} = 3 = -a_{12}$ ✓
All diagonal elements are 0 ✓
Answer: Skew-symmetric matrix
Prove that every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
Solution:
Let $A$ be any square matrix. We can write:
$$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$Let $P = \frac{A + A^T}{2}$ and $Q = \frac{A - A^T}{2}$
Check if $P$ is symmetric:
$$P^T = \left(\frac{A + A^T}{2}\right)^T = \frac{A^T + (A^T)^T}{2} = \frac{A^T + A}{2} = P$$✓
Check if $Q$ is skew-symmetric:
$$Q^T = \left(\frac{A - A^T}{2}\right)^T = \frac{A^T - (A^T)^T}{2} = \frac{A^T - A}{2} = -\frac{A - A^T}{2} = -Q$$✓
Therefore: $A = P + Q$ where $P$ is symmetric and $Q$ is skew-symmetric.
Uniqueness: Suppose $A = P_1 + Q_1 = P_2 + Q_2$. Then: $P_1 - P_2 = Q_2 - Q_1$
Left side is symmetric, right side is skew-symmetric. The only matrix that is both symmetric and skew-symmetric is the zero matrix.
Therefore: $P_1 = P_2$ and $Q_1 = Q_2$ Proved!
Quick Revision Box
| Concept | Key Point |
|---|---|
| Matrix order | $m \times n$ = rows × columns |
| Element notation | $a_{ij}$ = row $i$, column $j$ (RC!) |
| Square matrix | $m = n$ |
| Diagonal matrix | $a_{ij} = 0$ for $i \neq j$ |
| Identity matrix | $I_n$ has diagonal = 1, rest = 0 |
| Symmetric | $A = A^T$ (mirror across diagonal) |
| Skew-symmetric | $A = -A^T$ (diagonal must be 0) |
| Trace | Sum of diagonal elements |
| Matrix equality | Same order + all elements equal |
Related Topics
Within Matrices & Determinants Chapter
- Matrix Algebra — Operations: addition, multiplication, transpose
- Determinants Evaluation — Finding determinant values
- Adjoint and Inverse — Inverse matrices using symmetric properties
Math Connections
- Linear Algebra — Vector spaces and linear transformations
- Coordinate Geometry — Transformation matrices for rotations
- Systems of Equations — Matrix representation of linear systems
- Complex Numbers — Matrices with complex entries
Real-World Applications
- Computer Graphics — Transformation matrices for 3D rotations
- Machine Learning — Neural network weight matrices
- Cryptography — Encryption using matrix operations
- Economics — Input-output models using matrices
Teacher’s Summary
- Matrix order is always rows × columns (remember “RC Cola”)
- Element notation $a_{ij}$: first index = row, second = column
- Square matrices ($n \times n$) are special — they can have determinants and inverses
- Diagonal matrices have zeros off the diagonal; diagonal elements can be anything
- Symmetric matrices satisfy $A = A^T$ (mirror property)
- Skew-symmetric matrices satisfy $A = -A^T$ and must have zero diagonal
- Identity matrix $I$ acts like the number 1 in multiplication
- Any square matrix = symmetric part + skew-symmetric part
“Master matrix types — they’re the building blocks for all matrix operations in JEE!”
Exam Strategy: For JEE, quickly identify matrix types by checking:
- Is it square? → Check for special types
- Compare $a_{ij}$ and $a_{ji}$ → Symmetric or skew-symmetric?
- Check diagonal → Diagonal/scalar/identity?