Elements of a Matrix - Definition, aij Notation, and Examples

Every Entry In A Matrix Has Its Own Street Address

A matrix looks like a wall of numbers, but not one of those numbers is anonymous. Each has an exact address - a row and a column - and the notation $a_{ij}$ is how you name it without pointing.

Read the subscript in order and the address is unambiguous: the first number is the row, the second is the column. Once that convention is fixed, you can talk about any single value in a matrix of any size, and you can write general rules that apply to every element at once.

What Is An Element Of A Matrix?

An element of a matrix (also called an entry) is a single value that sits at one position in the matrix. A matrix is a rectangular arrangement of such values in rows and columns, and each value inside it is one element.

Consider the matrix

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

Every one of the nine numbers, $4$, $3$, $2$, $1$, $5$, $6$, $7$, $8$, $9$, is an element of $A$. Elements are not limited to plain numbers; they can be variables or algebraic expressions, so

$$B = \begin{bmatrix} x & 2y \ x+1 & 0 \end{bmatrix}$$

has the four elements $x$, $2y$, $x+1$, and $0$. What makes something an element is not its type but its position: one value, one cell. The grid that holds them is set by the order of a matrix, which fixes how many rows and columns there are.

What Is The aij Notation For Matrix Elements?

Each element of a matrix $A$ is named by a lowercase letter with two subscripts:

$$a_{ij} = \text{the element in row } i \text{ and column } j.$$

The first subscript is always the row; the second is always the column. So in the matrix $A$ above:

$$a_{11} = 4, \qquad a_{23} = 6, \qquad a_{32} = 8.$$

Reading $a_{23}$: go to row $2$, then column $3$ - the value there is $6$. The whole matrix can then be written compactly as $A = [a_{ij}]$, which says "$A$ is the matrix whose entry in row $i$, column $j$ is $a_{ij}$" for all valid $i$ and $j$. That single piece of notation is what lets textbooks state rules like "the transpose swaps $a_{ij}$ with $a_{ji}$" without drawing a single grid - the same shorthand that powers the transpose of a matrix and every operation built on entries.

How Do You Find The Position Of An Element In A Matrix?

To locate an element, read its subscript row-first, column-second, and count from the top-left corner.

Take the matrix

$$M = \begin{bmatrix} 10 & 20 & 30 & 40 \ 50 & 60 & 70 & 80 \end{bmatrix}.$$

• $m_{13}$ is row $1$, column $3$, so $m_{13} = 30$.

• $m_{24}$ is row $2$, column $4$, so $m_{24} = 80$.

• $m_{21}$ is row $2$, column $1$, so $m_{21} = 50$.

The order of the two subscripts is not decorative. $m_{13}$ and $m_{31}$ point to different cells - and in a matrix that is not square, $m_{31}$ might not even exist, since $M$ here has only two rows. Row first, column second, every time.

How Many Elements Does A Matrix Have?

The total number of elements equals the product of the number of rows and columns. For a matrix of order $m \times n$:

$$\text{number of elements} = m \times n.$$

• A $3 \times 3$ matrix has $3 \times 3 = 9$ elements.

• A $2 \times 4$ matrix has $2 \times 4 = 8$ elements.

• A $5 \times 1$ column matrix has $5 \times 1 = 5$ elements.

There is a neat consequence: a square matrix (equal rows and columns) always has a perfect-square number of elements, $1, 4, 9, 16, \dots$, while a matrix whose element count is not a perfect square can never be square. Counting the entries is also the first step to knowing how many equations you are really dealing with when you solve matrices in a linear system.

What Are Diagonal And Off-Diagonal Elements?

Within a square matrix, the elements split into two natural groups based on their address.

• Diagonal elements are those where the row index equals the column index - the entries $a_{ii}$, running from the top-left to the bottom-right. In the matrix $A$ above, the diagonal elements are $a_{11} = 4$, $a_{22} = 5$, and $a_{33} = 9$.

• Off-diagonal elements are all the rest, where $i \neq j$.

The set of diagonal positions for a $3 \times 3$ matrix is ${(1,1), (2,2), (3,3)}$. This split matters because whole families of matrices are defined by it: a diagonal matrix is one whose off-diagonal elements are all $0$, and an identity matrix is a diagonal matrix whose diagonal elements are all $1$.

When Are Two Matrices Equal, Element By Element?

Two matrices are equal only when they have the same order and every corresponding element matches:

$$A = B \iff a_{ij} = b_{ij} \text{ for all } i, j.$$

This element-wise rule is what lets you solve for unknowns hidden inside a matrix. If

$$\begin{bmatrix} x & 3 \ 5 & y \end{bmatrix} = \begin{bmatrix} 2 & 3 \ 5 & 7 \end{bmatrix},$$

then matching positions forces $x = 2$ and $y = 7$. Equality is not "roughly the same shape"; it is a strict, position-by-position agreement.

Examples Of Elements of a Matrix

The set runs from listing entries, through the index-order mistake, up to counting, diagonal elements, and solving an equal-matrix problem.

Example 1

List all the elements of $P = \begin{bmatrix} 6 & -1 \ 0 & 9 \end{bmatrix}$ and name each by its address.

Read across each row, top to bottom.

$$p_{11} = 6, \quad p_{12} = -1$$ $$p_{21} = 0, \quad p_{22} = 9$$

Final answer: the four elements are $6, -1, 0, 9$, at positions $(1,1), (1,2), (2,1), (2,2)$.

Example 2

Find the element $a_{31}$ of $A = \begin{bmatrix} 4 & 3 & 2 \ 1 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$.

Wrong path. A student reads $a_{31}$ as "column $3$, row $1$" and answers $2$. The subscript order was reversed. In a matrix the two numbers are not interchangeable, and swapping them lands you in a different cell entirely.

Correct. Read row first, column second.

$$a_{31} = \text{row } 3, \text{ column } 1$$ $$a_{31} = 7$$

Final answer: $a_{31} = 7$. The element $a_{13}$ (row $1$, column $3$) is the different value $2$; the order of the subscripts is the whole point.

Example 3

How many elements are in a matrix of order $4 \times 3$?

Multiply rows by columns.

$$\text{number of elements} = 4 \times 3 = 12$$

Final answer: $12$. Because $12$ is not a perfect square, this matrix cannot be square.

Example 4

Identify the diagonal elements of $C = \begin{bmatrix} 2 & 7 & 1 \ 4 & 5 & 9 \ 3 & 8 & 6 \end{bmatrix}$.

Diagonal elements are those with equal row and column indices, $c_{ii}$.

$$c_{11} = 2, \quad c_{22} = 5, \quad c_{33} = 6$$

Final answer: the diagonal elements are ${2, 5, 6}$. Every other entry is off-diagonal, where $i \neq j$.

Example 5

A matrix $A = [a_{ij}]$ of order $2 \times 2$ is defined by the rule $a_{ij} = i + j$. Write out $A$.

Substitute each position into the rule, one element at a time.

$$a_{11} = 1 + 1 = 2$$ $$a_{12} = 1 + 2 = 3$$ $$a_{21} = 2 + 1 = 3$$ $$a_{22} = 2 + 2 = 4$$

Assemble the grid.

$$A = \begin{bmatrix} 2 & 3 \ 3 & 4 \end{bmatrix}$$

Final answer: $A = \begin{bmatrix} 2 & 3 \ 3 & 4 \end{bmatrix}$. A rule for $a_{ij}$ builds the whole matrix, one address at a time.

Example 6

Find $x$ and $y$ given $\begin{bmatrix} x+1 & 8 \ 6 & 2y \end{bmatrix} = \begin{bmatrix} 5 & 8 \ 6 & 10 \end{bmatrix}$.

Equal matrices agree element by element, so match the positions that contain unknowns.

$$x + 1 = 5 \implies x = 4$$ $$2y = 10 \implies y = 5$$

Final answer: $x = 4$ and $y = 5$. Matrix equality turns one matrix statement into a set of simple element equations.

Why Naming Every Entry Was A Breakthrough

"How do you write one rule that speaks to thousands of numbers at once?"

Before matrices had this indexed notation, systems of equations were written out in full, line after line, with no compact way to refer to "the coefficient in the third equation, second variable." The English mathematician Arthur Cayley (1821–1895) developed the theory of matrices as objects in their own right in the 1850s, building on the term "matrix" that his colleague James Sylvester had coined. The move that made it all work was exactly the $a_{ij}$ address: once every entry has a row-column name, an operation on a whole matrix can be written as a single rule on a general element.

That is why the humble subscript matters far beyond the classroom. When a computer stores an image, each pixel is an element $a_{ij}$ of a grid; when a search engine ranks pages, the link structure is a giant matrix whose entries are addressed the same way; when a spreadsheet computes across a table, "cell $B3$" is nothing but $a_{ij}$ in disguise. Learning to read $a_{23}$ cleanly is the first rung of a ladder that reaches into matrices and determinants and every field that stores data in grids.

Common Mistakes

Mistake 1: Reversing The Row And Column Subscripts

Where it slips in: reading any element $a_{ij}$ where $i \neq j$.

Don't do this: interpret $a_{31}$ as "column $3$, row $1$." The first subscript is the row, always.

The correct way: read row first, column second. The rusher who skims the subscript is the one who returns $a_{13}$'s value when asked for $a_{31}$, and in a non-square matrix, the reversed address may not even exist.

Mistake 2: Confusing The Number Of Elements With The Order

Where it slips in: describing a matrix's size.

Don't do this: say a $2 \times 3$ matrix "has order $6$." Order is the pair $m \times n$; $6$ is the count of elements.

The correct way: keep the two ideas separate — order is $2 \times 3$ (rows by columns), and the number of elements is the product $2 \times 3 = 6$. The memoriser who blends them loses the shape information that the order carries.

Mistake 3: Treating Diagonal Elements As "The Bottom-Left To Top-Right" Line

Where it slips in: identifying diagonal elements.

Don't do this: pick the anti-diagonal (bottom-left to top-right) and call those the diagonal elements.

The correct way: the main diagonal runs top-left to bottom-right, the entries $a_{ii}$ where the row and column indices are equal. The other corner-to-corner line is the anti-diagonal, and it is a different set.

Conclusion

• The elements of a matrix are its individual entries, each at one row-column position.

• The notation $a_{ij}$ names the element in row $i$, column $j$ — row first, column second, always.

• A matrix of order $m \times n$ has $m \times n$ elements; square matrices have a perfect-square count.

• Diagonal elements are the $a_{ii}$ from top-left to bottom-right; off-diagonal elements have $i \neq j$.

• Two matrices are equal only when every corresponding element matches, which is how you solve for unknowns inside a matrix.

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Read More

• Multiplication of matrices — how elements combine when two matrices multiply.

• Determinant of a matrix — the single number built from a square matrix's elements.

• Cofactor matrix — replacing each element with its signed minor.

• Symmetric matrix — matrices where a_ij equals a_ji across the diagonal.

• Adjoint of a matrix — the transposed cofactor grid used to build inverses.