Matrix definition and notation

Open any spreadsheet of student marks — rows for students, columns for subjects — and you are already looking at a matrix: a rectangular block of numbers where position carries meaning, not just the number itself. Move a mark to the wrong row and it becomes someone else's score, even though the numbers on the page haven't changed. This opening idea of the CBSE Class 12 Matrices chapter gives that block of numbers a precise name, a precise way to write it down, and a precise way to point at any single entry inside it. Every later operation — addition, multiplication, inversion — depends on this notation being second nature, because those operations are defined entry-by-entry using exactly the labels introduced here. Get the row-then-column convention wrong at this stage and every later calculation inherits the mistake.

The core insight

A matrix is a rectangular array of $m \times n$ numbers, and its notation $A = [a_{ij}]_{m \times n}$ locates every entry uniquely by row index $i$ then column index $j$ .

The core idea

A matrix is a rectangular arrangement of numbers — called elements or entries — set out in rows (horizontal lines) and columns (vertical lines), and enclosed in brackets. If it has $m$ rows and $n$ columns, it is called a matrix of order $m \times n$ , read " $m$ by $n$ ," and written in full as

A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}

where: $a_{ij}$ is the entry sitting in row $i$ and column $j$ , $m$ is the total number of rows, and $n$ is the total number of columns. Matrices themselves are named with capital letters ( $A$ , $B$ , $C$ , ...) while their individual entries take the matching small letter with two subscripts. The order of those subscripts is a strict convention: the row index comes first, the column index second — never the reverse. A shorthand for the whole array is $A = [a_{ij}]_{m \times n}$ , read as "the matrix whose $(i,j)$ entry is $a_{ij}$ , having $m$ rows and $n$ columns." This compact form is how matrices are described whenever the entries follow a rule rather than being listed by hand. Within this syllabus, entries are real numbers and matrices used for computation go up to order $3 \times 3$ . Square brackets are the CBSE standard, but round brackets $(\ )$ mean exactly the same thing and are accepted in examinations. What is never interchangeable is a matrix with a determinant: a matrix is an array with no single numerical value, while a determinant (met in the next chapter) is a specific number computed from a square matrix. The figure below fixes how the two indices read off a position.

Figure

A 3 by 3 grid showing rows numbered 1 to 3 running across and columns numbered 1 to 3 running down, with the entry in row 2 column 3 highlighted and labelled a two three, row index first then column index

Derivation AConstructing a matrix from a stated general term

The goal is to show, with nothing beyond substitution and careful bookkeeping of two independent indices, exactly how a compact rule like $A = [a_{ij}]_{m \times n}$ with a formula for $a_{ij}$ expands into the full written-out array — the single mechanical skill this entire opening topic is built to test, and the one step every later matrix operation silently assumes you can already do.

Step 1. Read off the order from the subscript on the bracket notation. If $A = [a_{ij}]_{m \times n}$ , then $A$ has $m$ rows and $n$ columns, so the row index $i$ must run through every integer from $1$ to $m$ , and, independently, the column index $j$ must run through every integer from $1$ to $n$ . The two indices vary independently of each other — fixing $i$ does not fix $j$ , and vice versa.

Step 2. Count how many entries must be produced. Since $i$ takes $m$ possible values and $j$ takes $n$ possible values, and every combination of an $i$ -value with a $j$ -value gives one distinct entry, the total number of entries in $A$ is $m \times n$ — exactly the order written in the subscript. This is why "order $m \times n$ " and " $m \times n$ entries" are the same statement seen from two angles.

Step 3. Substitute each $(i,j)$ pair into the given rule for $a_{ij}$ to generate the actual numbers, working systematically — row by row, or column by column — so that no pair is skipped or repeated. Each substitution is independent of the others: the value of $a_{21}$ never depends on what $a_{12}$ turned out to be.

Step 4. Assemble the computed numbers back into the rectangular array, keeping the row index fixed as you move left to right along a row, and the column index fixed as you move top to bottom down a column. Placing a value in the wrong cell — even if every individual computation was correct — produces a genuinely different matrix, because a matrix is defined by both its entries and their arrangement.

Worked example

For the matrix

B = \begin{bmatrix} 7 & -2 & 0 \\ 5 & 4 & 9 \end{bmatrix}

state the order of $B$ , identify the entries $b_{12}$ , $b_{23}$ , and $b_{21}$ , and give the total number of entries.

Step 1. Count rows and columns directly from the array: $B$ has 2 horizontal rows and 3 vertical columns, so its order is $2 \times 3$ , written $B = [b_{ij}]_{2 \times 3}$ .

Step 2. Locate $b_{12}$ : row 1, column 2, reading across the top row to the second position, which is $-2$ .

Step 3. Locate $b_{23}$ : row 2, column 3, reading across the bottom row to the third position, which is $9$ .

Step 4. Locate $b_{21}$ : row 2, column 1, the first entry of the bottom row, which is $5$ .

Step 5. Apply the counting result from the derivation: total entries $= m \times n = 2 \times 3 = 6$ , matching the six numbers actually written in $B$ .

Answer: $B$ has order $2 \times 3$ with $6$ entries; $b_{12} = -2$ , $b_{23} = 9$ , and $b_{21} = 5$ .

Competitive stretch

If a matrix $C = [c_{ij}]_{4 \times 3}$ is defined by $c_{ij} = i + j$ , find $c_{31}$ and the total number of entries in $C$ without writing out the whole array. Step 1. Substitute directly: $c_{31} = 3 + 1 = 4$ — no need to build the other 11 entries first. Step 2. Total entries $= m \times n = 4 \times 3 = 12$ . Answer: $c_{31} = 4$ , and $C$ has $12$ entries.

Common traps

Swapping the subscript order — $a_{ij}$ always means row $i$ first, column $j$ second; $a_{23}$ and $a_{32}$ are generally different numbers.
Writing $|A|$ (a determinant, a single number) when the question asks about the matrix $A$ itself (an array with no numerical value) — a very common exam slip.
Miscounting rows and columns on a multi-line printed matrix — count horizontally for rows, vertically for columns, and double-check against the stated order.
Assuming a matrix must be square; rectangular matrices with $m \neq n$ , like the $2 \times 3$ example above, are just as valid.

Ask the tutor: Want to practise building a matrix from a general-term formula, or check whether two matrices you've written are actually equal? Ask the AI tutor to work through a few more with you.

Why does the row index always come before the column index — is it just a convention we have to memorise?

Yes, it is a fixed convention, but it is not arbitrary — it matches how you already read a table: you say which row you are in before saying which column, the same way you'd say "second shelf, third book" rather than the reverse. Because every later definition (matrix equality, addition, multiplication) is built entry-by-entry using $a_{ij}$ , keeping row-then-column consistent is what lets those definitions work without ambiguity. Practise reading a few printed matrices aloud, row first, until it feels automatic.

Are square brackets and round brackets really interchangeable for writing a matrix, or does CBSE prefer one?

Both $[\ ]$ and $(\ )$ denote a matrix and CBSE accepts either in an examination answer. Square brackets are more common in Indian textbooks and are what NCERT uses, so it is sensible to default to them in your own written work, but you will not lose marks for using parentheses as long as the rows and columns are clearly aligned inside them.

If two matrices have exactly the same numbers, are they always the same matrix?

Not necessarily. A matrix is defined by both its order and the exact position of every entry, so $\begin{bmatrix}1 & 2\end{bmatrix}$ and $\begin{bmatrix}1 \\ 2\end{bmatrix}$ contain the same two numbers but are different matrices — one is $1 \times 2$ , the other $2 \times 1$ . Even two matrices of the same order are different unless every entry matches in the same position; this idea is developed fully as the equality of matrices later in the chapter.

Matrix definition and notation

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads