← Reasoning

Figure Matrix

Reasoning · Non-Verbal Reasoning

6 Part2 worked examples5 practice questions

A matrix is a table of cells. Here means three rows and three columns, with one figure in each cell. Eight cells are filled; the ninth (bottom right) is empty. Every row and every column follows one and the same rule. Your job is to find the rule and build the figure of the empty cell.

Two kinds of matrix come up:

KindWhat is in a cellRule
QualitiesSome shapes — kind, number, shadingA separate rule for each quality
LinesA group of linesAdding, removing or sharing the lines of two cells

How to use this chapter

  1. In Part 1 learn to read a matrix and the order of checks.
  2. In Part 2 do the quality matrices and in Part 3 the line matrices.
  3. In each question the matrix is at the top (orange frame) and the options A–D below. Find the rule yourself, then choose the option label below. Open the solution last.
AssuranceEvery question was checked by computer: all the possible rules that fit the eight visible cells give the same empty figure. So every question has a single answer and no rival rule can give a different one.
1

Foundation — how to read a matrix

Read the rows, read the columns, test the rule

The order of finding the rule in a matrix and the kinds of rule.

Reading rows and columns

Separate the qualities and test each one alone

Guessing the rule from the first row and testing it on the second row.

Do not look at the third row first — it holds the empty cell. Guess the rule from the first row and test it on the second row. Then do the same with the columns. The rule that holds in both the first and the second row is the one to apply to the third row.

The order of reading a matrix
1. Split the figure into qualities (kind, number, shading / lines)2. Look at each quality along a row3. Test the same rule on the second row and on the columns4. Work out each quality for the empty cell
TrickConcentrate on one quality and forget the rest. The eye cannot hold three qualities at once. A quality that never changes anywhere in the matrix is constant; set it aside.
?(A)(B)(C)(D)
Question figure and options
Example 1 · Basic
In the matrix above every row and column follows a rule and the last cell is empty (?). Which figure completes it?
Solution, step by step
  1. 1Split the figures into three qualities — kind, number, shading — and find the rule of each quality separately.
  2. 2Kind of shape: the three values appear once in every row and every column (a Latin square). So the last cell holds circles.
  3. 3Number: the same in every cell. So 2 of them.
  4. 4Shading: the same in every cell. So half-shaded (orange).
  5. 5Putting the three together gives option (C); in each of the other options one quality is wrong.
Answer Figure C
Test the rule of each quality separately — only the kind of shape first, then the number, then the shading.
?(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
In the matrix above every row and column follows a rule and the last cell is empty (?). Which figure completes it?
?(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
In the matrix above every row and column follows a rule and the last cell is empty (?). Which figure completes it?
?(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
In the matrix above every row and column follows a rule and the last cell is empty (?). Which figure completes it?

The kinds of rule

Constant, changing in order, Latin square, adding and taking away

A table of all the rules that come up in matrices.

RuleWhat it looks likeFor the empty cell
ConstantThe same all over the matrixThe same value
Row-wise / column-wiseAlike in a row (column), different in the nextThe value of its row (column)
Latin squareThree values once in every row and columnThe third value left over
Growing / falling in orderNumber 1, 2, 3 or 3, 2, 1The next in order
Adding / subtractingThird cell = first ± secondThe same sum in the last row
Combining linesAdding, removing the common, keeping the common, taking awayThe same action in the last row

The last two kinds make the third cell out of the first two. In a number matrix test the counts (first + second = third?); in a line matrix look at how the lines combine.

Remember: one rule runs over the whole matrix. If you see different rules on two rows, it is your reading that is wrong, not the matrix.

?(A)(B)(C)(D)
Question figure and options
Example 2 · Basic
In the matrix above every cell is made of lines and the last cell is empty (?). Which figure completes it?
Solution, step by step
  1. 1To find the rule look at the first row: cell 3 is made from cells 1 and 2.
  2. 2For example: the first cell has the right line, the slanting line (\), the vertical mid-line; the second the left line, the slanting line (/), the vertical mid-line; the third the left line, the right line, the slanting line (\), the slanting line (/), the vertical mid-line.
  3. 3That is the rule — adding: all the lines of both cells. Check it on the second row too: it fits there as well.
  4. 4Applied to the last row: the left line, the slanting line (/), the vertical mid-line, the horizontal mid-line — that is option (B).
  5. 5Each of the other options has a line too many or too few, or is the result of a rival rule.
Answer Figure B
In the first row see which lines survive into the third cell and which are dropped.
?(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
In the matrix above every cell is made of lines and the last cell is empty (?). Which figure completes it?
?(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
In the matrix above every row and column follows a rule and the last cell is empty (?). Which figure completes it?
2

Matrices of qualities

Kind of shape, number and shading

Some shapes in each cell — finding three rules, for kind, number and shading, one by one.

The kind of shape and its shading

Latin square, row-wise and chessboard

When the number is constant and only the shape or the shading changes.

🔒

The number of shapes

Growing in order, sum, difference and equal totals

The rules for the number: growing or falling, third = first ± second, and equal row totals.

🔒

All three qualities together

A separate rule for each quality, then put the answer together

Solving each quality alone when kind, number and shading all change.

🔒
3

Matrices of lines

Making the third cell from two

Adding lines, removing the common ones, keeping the common ones and taking away.

Adding, and removing the common lines

All the lines versus only the lone lines

The two commonest line rules: adding and the “difference” (a line in both cells vanishes).

🔒

Taking away, and keeping the common

First minus second, or keep only the common

Three more line rules and how they differ from adding.

🔒
4

Strategy guide

The order of checks and the slips to avoid

The order of checks in any matrix question, common traps and time-savers.

The order of checks

Split qualities → row → column → answer

Which rule to try first on seeing a matrix.

🔒

Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in matrices.

🔒
5

Mixed practice

All kinds mixed — first spot the kind of matrix

Matrices of qualities and of lines in three levels.

Basic practice

One or two qualities change

Simple Latin squares, numbers growing in order and the adding rule for lines.

🔒

Standard practice

All three qualities, all five line actions

Sum-and-difference numbers, Latin squares and every line action.

🔒

Advanced practice

Dense matrices, mixed rules

Three-quality matrices and line actions with more lines.

🔒
6

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

🔒