Year 12 Discrete Mathematics: Rules for matrix elements

Matrix-element questions focus on structure. Instead of treating a matrix as a random rectangle of numbers, you are asked to recognise how each entry depends on its row and column position. This is one of the cleanest ways to see what discrete mathematics is doing: it gives a rule for a finite grid of values and asks you to interpret the pattern. Once you understand what an entry like $a_{i j}$ means, many matrix questions become far more systematic.

The notation

a_{i j}

means the entry in row

i

, column

j

. Keep row and column meaning separate all the way through.

Subtopic 1: Using a position rule to build a matrix

A position rule gives a formula for the entry in terms of the row number and column number. The job is then to substitute the appropriate values of $i$ and $j$ into that rule. These questions are simple once the notation is stable, but students often reverse row and column or lose track of which index they are substituting.

Worked example 1

Problem: A matrix

A

has entries

a_{i j} = 2 i + j

. Find the

2 x 3

matrix.

For row $i = 1$ : $a_{11} = 2 (1) + 1 = 3$ , $a_{12} = 4$ , $a_{13} = 5$ .
For row $i = 2$ : $a_{21} = 5$ , $a_{22} = 6$ , $a_{23} = 7$ .

Answer:

A = [[3, 4, 5], [5, 6, 7]]

Subtopic 2: Identifying entries from a given matrix

Some questions go in the other direction: they give the matrix and ask you to identify a particular element, compare two entries, or infer the pattern. In those cases the meaning of the subscripts is again the whole issue. If the first subscript is the row and the second is the column, then $a_{23}$ is not interchangeable with $a_{32}$ . The structure of the grid matters.

Worked example 2

Problem: Let

B = [[1, 4, 7], [2, 5, 8], [3, 6, 9]]

. Find

b_{31}

and

b_{23}

$b_{31}$ means row 3, column 1, so the value is $3$ .
$b_{23}$ means row 2, column 3, so the value is $8$ .

Answer:

b_{31} = 3

and

b_{23} = 8

Why matrix-entry rules matter

Matrix-element rules train a very important discrete-mathematics habit: mapping position to meaning. In later modelling questions, rows and columns may represent states, age groups, locations, or players. If you already understand how entries are indexed and interpreted, those later questions become much more manageable. The arithmetic is rarely the main difficulty. The main difficulty is reading the structure correctly and preserving it through the calculation.

Worked example 3

Problem: A matrix

C

is defined by

c_{i j} = i - j

. Find

c_{12}

c_{21}

, and explain why they differ.

$c_{12} = 1 - 2 = - 1$ .
$c_{21} = 2 - 1 = 1$ .
They differ because switching row and column changes the expression itself.

Answer:

c_{12} = - 1

and

c_{21} = 1

, because row and column positions are not interchangeable.

Common traps

Reversing row and column subscripts.
Applying the position rule with the wrong values of $i$ and $j$ .
Assuming symmetric behaviour when the entry rule is not symmetric.
Writing a matrix of the wrong size because the dimensions were ignored.

Revision focus

When revising, say each subscripted entry aloud as "row $i$ , column $j$ " before calculating. This makes the indexing much harder to reverse accidentally. It also helps to sketch the matrix grid and label the rows and columns if the rule feels abstract.

Another useful strategy is to test a pattern on just two or three entries before building the whole matrix. If those first entries agree with the rule, the rest of the construction is much safer. In discrete mathematics, confirming the structure early often prevents a lot of wasted calculation later.

Practice links

Rules for matrix elements