Matrix reasoning (3x3 grids) - Year selective Numerical Reasoning

What you will learn

read a 3x3 matrix and identify whether the rule runs along rows, down columns, or both,
try the common rule families: constant sum, constant product, equal ratios, or “left + middle = right”,
work diagonally when row and column rules both fail,
use a complete row to test a candidate rule before applying it to the row with the gap,
eliminate options quickly by sanity-checking parity (odd/even) and size.

Worked example 0 A sum matrix

Grid:

 2   5   7
 3   4   7
 6   1   ?

Step 1. Look at row 1: is $2 + 5 = 7$ ? Yes.

Step 2. Check row 2: $3 + 4 = 7$ . Yes.

Step 3. Rule confirmed: right column = left + middle. Row 3: $6 + 1 = 7$ . Answer: 7.

1. Row rules: sums, differences, products

Most “easy” matrices have the third column equal to a simple combination of the first two.

Worked example 1 Product rule

 2   3   6
 4   2   8
 5   3   ?

Row 1: $2 \times 3 = 6$ . Row 2: $4 \times 2 = 8$ . Rule: right = left * middle. Row 3: $5 \times 3 = 15$ . Answer: 15.

2. Column rules

If rows don’t give a clean pattern, each column may have its own pattern going down.

Worked example 2 Each column adds up

 1   2   3
 4   5   6
 5   7   ?

Column 1: $1 + 4 = 5$ . Column 2: $2 + 5 = 7$ . Column 3: $3 + 6 = 9$ . Rule: bottom row = top + middle. Answer: 9.

3. Positional rules (every cell from its row and column number)

Sometimes the value in each cell depends on its position in a systematic way.

Worked example 3 Row * column

If row 1 and column 1 are both labelled 1, and the value in each cell is (row number + column number):

 2   3   4
 3   4   5
 4   5   ?

Cell at row 3, column 3: $3 + 3 = 6$ . Answer: 6. (Good sanity check: every diagonal cell is $2 \times r$ , so the bottom-right corner would be $6$ .)

4. Mixed / two-rule matrices

One rule runs along rows, and a different rule runs along columns. Check both.

Worked example 4 Rows double, columns add 1

 1   2   4
 2   4   8
 3   6   ?

Rows: each term doubles going right ( $1 \to 2 \to 4$ ; $2 \to 4 \to 8$ ). Row 3 continues: $3 \to 6 \to 12$ . Answer: 12. Column check: column 3 reads 4, 8, 12 — gaps of 4, consistent.

Drill pack

10 minutes. For each grid, find the missing value. The grids are written as three rows of three values. ”?” is the missing cell.

Drill timer

Count down

00:00

Target: 10:00

Previous attempts

No attempts yet.

Target: 10 minutes. Time yourself with a watch or phone timer — no pausing.

Fluency

Timed drill

Rows $[1, 2, 3]$ , $[4, 5, 6]$ , $[7, 8, ?]$ . Rule: each row increases by 1. Find ?. (a) 8 (b) 9 (c) 10 (d) 11
Rows $[2, 3, 5]$ , $[4, 5, 9]$ , $[6, 7, ?]$ . Right = left + middle. (a) 12 (b) 13 (c) 14 (d) 15
Rows $[1, 2, 2]$ , $[3, 4, 12]$ , $[2, 5, ?]$ . Right = left $\times$ middle. (a) 7 (b) 9 (c) 10 (d) 12
Rows $[10, 3, 7]$ , $[8, 2, 6]$ , $[9, 4, ?]$ . Right = left - middle. (a) 3 (b) 4 (c) 5 (d) 6
Rows $[2, 4, 8]$ , $[3, 6, 12]$ , $[4, 8, ?]$ . Each row doubles. (a) 12 (b) 14 (c) 16 (d) 20
Rows $[1, 4, 9]$ , $[2, 5, 10]$ , $[3, 6, ?]$ . Each column goes up by 1 going down. (a) 9 (b) 10 (c) 11 (d) 12
Rows $[5, 7, 9]$ , $[8, 10, 12]$ , $[6, 8, ?]$ . Each row gap +2. (a) 9 (b) 10 (c) 11 (d) 12
Rows $[12, 8, 4]$ , $[15, 10, 5]$ , $[18, 12, ?]$ . Each row is arithmetic; row 1 gap -4, row 2 gap -5, row 3 gap -6. (a) 4 (b) 6 (c) 8 (d) 9
Rows $[2, 3, 6]$ , $[4, 1, 4]$ , $[5, 2, ?]$ . Right = left $\times$ middle. (a) 8 (b) 9 (c) 10 (d) 15
Rows $[3, 3, 9]$ , $[4, 2, 8]$ , $[5, 2, ?]$ . Right = left $\times$ middle. (a) 8 (b) 9 (c) 10 (d) 12
Rows $[2, 1, 3]$ , $[5, 2, 7]$ , $[8, 4, ?]$ . Right = left + middle. (a) 10 (b) 11 (c) 12 (d) 13
Rows $[7, 4, 3]$ , $[9, 5, 4]$ , $[12, 8, ?]$ . Right = left - middle. (a) 3 (b) 4 (c) 5 (d) 6
Rows $[1, 2, 4]$ , $[2, 3, 5]$ , $[3, 4, ?]$ . Columns each add down. (a) 5 (b) 6 (c) 7 (d) 8
Rows $[10, 5, 2]$ , $[20, 10, 2]$ , $[30, 15, ?]$ . Right = left $\div$ middle. (a) 2 (b) 3 (c) 4 (d) 5
Rows $[1, 1, 2]$ , $[2, 3, 5]$ , $[3, 5, ?]$ . Right = left + middle. (a) 7 (b) 8 (c) 9 (d) 10
Rows $[4, 9, 13]$ , $[6, 7, 13]$ , $[8, 5, ?]$ . Right = left + middle. (a) 11 (b) 12 (c) 13 (d) 14
Rows $[25, 5, 5]$ , $[36, 6, 6]$ , $[49, 7, ?]$ . Middle = $\sqrt{\text{left}}$ , right = middle. (a) 6 (b) 7 (c) 8 (d) 9
Rows $[1, 3, 2]$ , $[4, 7, 3]$ , $[6, 10, ?]$ . Right = middle - left. (a) 2 (b) 3 (c) 4 (d) 5
Rows $[2, 6, 12]$ , $[3, 9, 27]$ , $[4, 12, ?]$ . Right = left $\times$ middle. (a) 36 (b) 40 (c) 44 (d) 48
Rows $[1, 4, 9]$ , $[2, 5, 10]$ , $[3, 6, ?]$ . Each column increases by 1 going down. (a) 9 (b) 10 (c) 11 (d) 12

Challenge

The rules mix operations or use column patterns only.

Reasoning

Harder patterns

Rows $[2, 3, 13]$ , $[1, 4, 17]$ , $[3, 2, ?]$ . Right = left $^2$ + middle $^2$ . (a) 11 (b) 13 (c) 14 (d) 17
Rows $[1, 2, 5]$ , $[3, 4, 25]$ , $[5, 6, ?]$ . Right = left $^2$ + middle $^2$ . (a) 49 (b) 51 (c) 61 (d) 55
Rows $[6, 2, 4]$ , $[10, 4, 6]$ , $[14, 6, ?]$ . Right = left - middle. (a) 6 (b) 7 (c) 8 (d) 9
Rows $[2, 4, 16]$ , $[3, 5, 25]$ , $[4, 6, ?]$ . Right = middle $^2$ . (a) 30 (b) 32 (c) 36 (d) 40
Rows $[12, 3, 4]$ , $[20, 4, 5]$ , $[30, 5, ?]$ . Right = left $\div$ middle. (a) 5 (b) 6 (c) 7 (d) 8
Rows $[1, 1, 1]$ , $[2, 4, 8]$ , $[3, 9, ?]$ . Column $n$ = (row) $^n$ . (a) 18 (b) 24 (c) 27 (d) 36
Rows $[5, 8, 13]$ , $[7, 9, 16]$ , $[6, 10, ?]$ . Right = left + middle. (a) 14 (b) 15 (c) 16 (d) 18
Rows $[2, 3, 7]$ , $[4, 1, 9]$ , $[5, 2, ?]$ . Right = $2 \times$ left + middle. (a) 11 (b) 12 (c) 13 (d) 14

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Each answer shows the rule (row, column, or both) that fits the two complete rows, then the value.

Timed drill

(b) 9. Each row counts up by 1: $7, 8, 9$ .
(b) 13. Right $=$ left $+$ middle. $6 + 7 = 13$ .
(c) 10. Right $=$ left $\times$ middle. $2 \times 5 = 10$ .
(c) 5. Right $=$ left $-$ middle. $9 - 4 = 5$ .
(c) 16. Each row doubles. $4, 8, 16$ .
(c) 11. Each column goes up by $1$ : column 3 reads $9, 10, 11$ .
(b) 10. Each row has gap $+2$ . $6, 8, 10$ .
(b) 6. Row 3 has gap $-6$ : $18, 12, 6$ .
(c) 10. Right $=$ left $\times$ middle. $5 \times 2 = 10$ .
(c) 10. Right $=$ left $\times$ middle. $5 \times 2 = 10$ .
(c) 12. Right $=$ left $+$ middle. $8 + 4 = 12$ .
(b) 4. Right $=$ left $-$ middle. $12 - 8 = 4$ .
(b) 6. Each column increases by $1$ going down: column 3 reads $4, 5, 6$ .
(a) 2. Right $=$ left $\div$ middle. $30 \div 15 = 2$ .
(b) 8. Right $=$ left $+$ middle. $3 + 5 = 8$ .
(c) 13. Right $=$ left $+$ middle. $8 + 5 = 13$ .
(b) 7. Middle $= \sqrt{\text{left}}$ , and right equals middle. $\sqrt{49} = 7$ .
(c) 4. Right $=$ middle $-$ left. $10 - 6 = 4$ .
(d) 48. Right $=$ left $\times$ middle. $4 \times 12 = 48$ .
(c) 11. Each column goes up by $1$ : column 3 reads $9, 10, 11$ .

Challenge

(b) 13. Right $=$ left $^2$ $+$ middle $^2$ . $3^2 + 2^2 = 9 + 4 = 13$ .
(c) 61. Right $=$ left $^2$ $+$ middle $^2$ . $5^2 + 6^2 = 25 + 36 = 61$ .
(c) 8. Right $=$ left $-$ middle. $14 - 6 = 8$ .
(c) 36. Right $=$ middle $^2$ . $6^2 = 36$ .
(b) 6. Right $=$ left $\div$ middle. $30 \div 5 = 6$ .
(c) 27. Column $n$ $=$ (row number) $^n$ . Row 3, column 3 $= 3^3 = 27$ .
(c) 16. Right $=$ left $+$ middle. $6 + 10 = 16$ .
(b) 12. Right $= 2 \times$ left $+$ middle. $2 \times 5 + 2 = 12$ .

Prefer paper? Print the answer key as a separate booklet: open print view ->