Element of a Matrix — Definition, Notation & Examples

Q: How do you read the subscript notation for a matrix element?

The notation $a_{ij}$ is read as 'a sub i j.' The first subscript $i$ always refers to the row, and the second subscript $j$ always refers to the column. For example, $a_{34}$ means the element in row 3, column 4.

Q: How many elements does a matrix have?

A matrix with $m$ rows and $n$ columns (called an $m \times n$ matrix) has exactly $m \times n$ elements. For example, a $3 \times 4$ matrix has 12 elements.

Element of a Matrix

One of the entries in a matrix. The address of an element is given by listing the row number then the column number.

Two 3x3 matrices showing notation: general matrix A with elements a₁₁ to a₃₃, and example matrix A = [3,0,-5; 2,-6,1; 4,-1,7].

Key Formula

a_{ij}

Where:

$a$ = The name of the matrix (often a capital letter like A, with lowercase for individual elements)
$i$ = The row number where the element is located (counted from top to bottom)
$j$ = The column number where the element is located (counted from left to right)

Worked Example

Problem: Given the matrix below, identify the element at row 2, column 3.

Step 1: Write out the matrix A.

A = \begin{bmatrix} 5 & 8 & 2 \\ 1 & 7 & 4 \\ 9 & 3 & 6 \end{bmatrix}

Step 2: Locate row 2. This is the second row from the top.

\text{Row 2: } \begin{bmatrix} 1 & 7 & 4 \end{bmatrix}

Step 3: Within row 2, locate column 3. This is the third entry from the left.

a_{23} = 4

Answer: The element at row 2, column 3 is

a_{23} = 4

Another Example

Problem: For the same matrix A, list all elements in column 1.

Step 1: Recall the matrix A.

A = \begin{bmatrix} 5 & 8 & 2 \\ 1 & 7 & 4 \\ 9 & 3 & 6 \end{bmatrix}

Step 2: Column 1 is the first column from the left. Read down the column to find each element.

a_{11} = 5, \quad a_{21} = 1, \quad a_{31} = 9

Answer: The elements in column 1 are 5, 1, and 9.

Frequently Asked Questions

How do you read the subscript notation for a matrix element?

The notation

a_{ij}

is read as 'a sub i j.' The first subscript

i

always refers to the row, and the second subscript

j

always refers to the column. For example,

a_{34}

means the element in row 3, column 4.

How many elements does a matrix have?

A matrix with

m

rows and

n

columns (called an

m \times n

matrix) has exactly

m \times n

elements. For example, a

3 \times 4

matrix has 12 elements.

Element vs. Entry

These two words mean exactly the same thing when referring to matrices. 'Element' and 'entry' are used interchangeably in textbooks and courses. Both refer to a single value at a specific row-column position within a matrix.

Why It Matters

Identifying individual elements is the foundation of nearly every matrix operation. When you multiply matrices, you compute each element of the result by combining specific rows and columns. In real-world applications like computer graphics, data science, and physics, accessing and manipulating individual matrix elements is how transformations, datasets, and systems of equations are handled.

Common Mistakes

Mistake: Reversing the order of subscripts — writing

a_{ji}

(column first, row second) instead of

a_{ij}

(row first, column second).

Correction: Always list the row number first and the column number second. A helpful memory aid: 'RC' — Row then Column, like the abbreviation for a remote control.

Mistake: Starting the count from 0 instead of 1.

Correction: In standard mathematical notation, matrix rows and columns are numbered starting from 1. The top-left element is

a_{11}

, not

a_{00}

. (Note: some programming languages do use 0-based indexing, but math convention starts at 1.)

Related Terms

Matrix — The rectangular array that contains the elements
Row of a Matrix — A horizontal line of elements in a matrix
Column of a Matrix — A vertical line of elements in a matrix
Dimensions of a Matrix — Row count times column count gives total elements
Matrix Multiplication — Each resulting element is computed from row-column pairs
Square Matrix — A matrix where row count equals column count