4.2 Number Matrices & Missing Values

Matrices as 2D pattern problems

A number matrix (or missing-number grid) shows numbers in rows and columns with one cell blank — or occasionally a value you must compute as the “result” of the grid. On the JOA, these items belong to numerical reasoning inside the mixed 51-question paper. They measure whether you can find a consistent rule across a small table, not whether you can prove linear algebra or solve simultaneous equations as a school topic.

Think of a matrix as several short number series glued together. The winning habit is the same classify–verify–answer loop, expanded to two dimensions.

Default search order (use this every time)

1. Complete rows — does each row share an add, multiply, or “outer cells produce middle” rule?
2. Complete columns — same checks vertically.
3. Same rule both ways — sometimes every row and every column obeys the same relationship (strong confirmation).
4. Diagonals / corners / centre special — only after simpler line rules fail.
5. Commit or flag — do not invent a one-off rule that works only on the incomplete line.

Reading a 3×3 grid

Label cells mentally:

A B C
D E F
G H I

If I is missing, you can use row 3 (G, H, I), column 3 (C, F, I), or a whole-grid rule that also uses A–H. Prefer rules already proven on rows 1–2 or columns 1–2.

Row rules that appear constantly

Worked example A — constant row sum

2  7  6
9  5  1
4  3  ?

Row sums: 2+7+6=15, 9+5+1=15, so 4+3+?=15 → ?=8. (This classic also has column sums of 15 — bonus confirmation.)

Worked example B — each row multiplies to the same product

2  3  6
1  4  4
2  2  ?

Products: 2×3×6=36, 1×4×4=16 — products are not equal, so abandon pure product. Instead notice middle × right = left? 3×6=18 ≠ 2. Try left × middle = right: 2×3=6, 1×4=4, 2×2=4. So ?=4.

Worked example C — third cell is sum or difference of the first two

4  9  13
7  2   9
8  5   ?

Each row: first + second = third. 8+5=13. Distractors might be 3 (difference) or 40 (product).

Worked example D — third cell is product-plus adjustment

3  4  14
2  5  12
4  3   ?

Test: first × second + 2 → 3×4+2=14, 2×5+2=12, 4×3+2=14. Hybrid row rules are common; keep the adjustment constant across rows when you claim it.

Column rules and “same rule both ways”

If rows look messy, rotate your attention down the columns without changing the numbers.

Worked example E — column arithmetic

5  2  7
3  8  4
2  ?  3

Suppose each column’s top − middle = bottom: column 1: 5−3=2 (matches), column 3: 7−4=3 (matches), column 2: 2−8=−6, so ?=−6 if negatives are allowed. If options are only positive, reject this rule and try another (for example middle = top + bottom: col1 3≠5+2, fails).

Always match the answer format: if every given cell is a positive integer and options are positive, a negative result is a warning that the rule is wrong — not necessarily that the JOA bans negatives, but that your rule is inconsistent with the presented style.

Worked example F — every row and column sums to 12

1  6  5
8  3  1
3  3  ?

Row1=12, row2=12 → row3: 3+3+?=12 → ?=6. Check columns: col1 1+8+3=12, col2 6+3+3=12, col3 5+1+6=12. Dual confirmation is ideal when time allows a 5-second check.

Diagonals, centres, and other whole-grid patterns

Use these when line-by-line sums/products fail.

Worked example G — centre is special

2  9  4
7  5  3
6  1  ?

One familiar pattern family (magic-style): all lines through the centre share a sum. Here 2+5+?= ?; 4+5+6=15; 2+5+8 would need ?=8 for diagonal; 6+5+4=15 already on the other diagonal pieces… Actually check: 2+9+4=15, 7+5+3=15, 6+1+8=15 if ?=8, and columns 2+7+6=15, 9+5+1=15, 4+3+8=15. Full magic-square style sum 15 works. You do not need the name “magic square” — you need the verified sum.

Worked example H — each row is a short series

2  4  8
3  6  12
4  8  ?

Each row doubles: ?=16. This is a matrix only in layout; the rule is a per-row geometric series. That is still valid numerical reasoning.

Worked example I — corners relate to the centre

6  1  8
7  5  3
2  9  ?

Suppose each pair of opposite corners sums with a fixed total through the centre idea, or try row sums again: 6+1+8=15, 7+5+3=15 → 2+9+?=15 → ?=4. Re-check columns before locking: 6+7+2=15, 1+5+9=15, 8+3+4=15. Again a uniform sum.

Teaching point: many “clever” diagonal stories collapse into a simpler line sum once you check. Prefer the simpler verified rule.

Typed numeric answers and official-style missing numbers

Official JOA familiarisation material shows that some numerical items may require a typed numeric response rather than clicking A–D. Process implications:

Worked typed-style example

10  2  12
 7  5  12
 9  4   ?

If each row’s first + second = third: 9+4=13. Type 13, not 5 (difference) or 36 (product). Under time pressure, people mis-add 9+4 as 12 because the previous rows both summed to 12 — a priming trap. Verify the rule definition (sum of first two) rather than copying the previous total blindly if the inputs changed.

Wait — in this grid both complete rows sum to 12 with first+second=third, and 9+4=13, so the third column is not a constant 12; the rule is still first+second=third. Do not force “all bottoms equal 12.”

Systematic checklist under ~25–35 seconds

1. Scan all given numbers for size range (small integers? growing fast?).
2. Test row sum / product / first∘second=third on every complete row.
3. If failed, test the same for columns.
4. If failed, try hybrid (× then +) on rows.
5. If failed, try “each row is a mini series.”
6. Only then diagonals / centre stories.
7. Apply the winning rule to the blank; spot-check one other complete line.
8. If still broken at ~35–40s, flag.

Common matrix traps

Mini grid drill

5  7  12
4  9  13
6  8   ?

First + second = third → ?=14.

2  5  10
3  4  12
4  4   ?

First × second = third → ?=16.

1  2  3
4  5  6
7  8  ?

Simple counting series across the grid → ?=9 (do not overcomplicate).

Matrix mastery is procedure under pressure: a fixed search order, ruthless verification on complete lines, and clean numeric entry when the interface asks you to type the missing value.