6.3 Matrix Reasoning

6.3 Matrix Reasoning

Quick Answer: A figure matrix is a grid where each row transforms the figure the same way and each column transforms the figure the same way. Find the row rule, find the column rule, and combine them to produce the missing cell. The most common error is to apply only the row rule and ignore the column rule.

Matrix reasoning (also called Raven-style matrices) is a staple of non-verbal intelligence tests, including the PAF GD Pilot Initial Test. A 3×3 grid of figures is shown with one cell blank. Your job is to infer the rule that governs the grid and select the figure that completes it.

The Two-Rule Structure

Every well-formed matrix has (at least) two independent rules:

1. A row rule — the way a figure changes as you move left-to-right across a row.
2. A column rule — the way a figure changes as you move top-to-bottom down a column.

The missing cell, which sits at the intersection of a particular row and a particular column, must satisfy both rules. If you find a figure that fits the row rule but not the column rule, it is a distractor.

The Derivation Method

1. Read the rows. Compare cell 1 to cell 2 to cell 3 in the first row. What changes? What stays the same? Write the rule in words ("the dot moves one position clockwise each step").
2. Check the rule on the other rows. If the rule holds for every complete row, you have the row rule. If not, revise.
3. Read the columns. Compare cell 1 of column 1 to cell 2 of column 1 to cell 3 of column 1. Derive the column rule in words.
4. Check the rule on the other columns. Confirm it holds.
5. Combine. Apply the row rule and the column rule together to generate the missing cell.

Worked Example 1 — Rotation Matrix

A 3×3 matrix shows the same L-shaped figure in every cell. Across each row, the figure rotates 90° CW from left to right. Down each column, the figure rotates 90° CW from top to bottom. The missing cell is the bottom-right.

• The top-left figure is at orientation 0°.
• The bottom-right cell is three rows down (3 × 90° = 270° CW) and three columns right (3 × 90° = 270° CW) from the top-left.
• Combined rotation: 270° + 270° = 540° = 180° (since 540° - 360° = 180°).

Answer: the L is rotated 180° from the top-left starting orientation.

Worked Example 2 — Element-Count Matrix

A 3×3 matrix shows stars in each cell. Row 1 has 1, 2, 3 stars. Row 2 has 2, 3, 4 stars. Row 3 has 3, 4, ? stars. The missing cell is the bottom-right.

• Row rule: each row adds one star as you move right (count increases by 1).
• Column rule: each column adds one star as you move down (count increases by 1).
• The bottom-right cell is in row 3, column 3. Row 3 already has 3, 4, so the next is 5. Column 3 has 3, 4, so the next is 5. Both rules agree.

Answer: 5 stars.

Worked Example 3 — Colour and Shape

A 3×3 matrix shows shapes that vary in two features: shape (circle, square, triangle) and shading (white, grey, black). Row 1: white circle, grey square, black triangle. Row 2: grey circle, black square, white triangle. Row 3: black circle, white square, ? triangle.

• Row rule: across a row, the shape cycles circle → square → triangle, and the shading cycles white → grey → black (with wrap-around).
• Column rule: down a column, the shape stays the same (column 1 is all circles, column 2 is all squares, column 3 is all triangles), and the shading cycles in the same order.
• The missing cell is in row 3, column 3, so the shape is triangle (column rule) and the shading follows the row wrap-around: row 3 is black, white, ? — the next after white is grey.

Answer: a grey triangle.

A Mermaid Diagram of the Two-Rule Method

The diagram shows how the row rule and column rule combine at the missing cell.

Common Traps

Trap 1 — Applying Only the Row Rule

This is the single most common error. Candidates read across the row containing the missing cell, derive a rule, and pick the first answer that fits. That answer almost always violates the column rule and is a planted distractor. Always check the column rule before committing.

Trap 2 — Additive vs Multiplicative Counts

When a matrix shows counts of elements (stars, dots, lines), the rule can be:

• Additive: counts increase by a fixed amount (e.g., +1 each step).
• Multiplicative: counts scale by a fixed factor (e.g., ×2 each step).

Test both. A row with 1, 2, 4 is multiplicative (×2), not additive (+1, +2). A row with 2, 4, 6 is additive (+2), not multiplicative.

Trap 3 — Ignoring a Third Feature

Some matrices vary three features at once (shape, shading, orientation). Candidates who track only two features will pick a distractor that matches those two but differs on the third. List every feature you can see before deriving rules.

Trap 4 — Wrap-Around Cycling

When a feature cycles (white → grey → black → white), the wrap-around is easy to miss. If row 2 is grey, black, white, the next after white is grey — not "nothing". Treat the cycle as circular.

A Chart of Common Rule Types

The chart below shows how often each rule type appears in a typical PAF non-verbal set. Rotation and element-count rules dominate; feature-combination rules are rarer but high-value because other candidates miss them.

Speed Strategy

• Scan the matrix for the most obvious feature first — usually the shape or the count. Derive its rule and you often get the answer in under 20 seconds.
• Use the diagonals as a check. In many matrices, the main diagonal carries a third rule (e.g., the shading is constant along the diagonal). If your row-and-column answer also satisfies the diagonal, you are almost certainly correct.
• Eliminate figures that violate any single feature. If the missing cell must be a triangle, eliminate every non-triangle option before reasoning about shading or orientation.
• Watch for the "no rule" trap. If a row has three identical cells, the row rule is "no change". Do not invent a rule where none exists.

Practice Drills

1. Build your own 3×3 matrix. Pick a figure and apply a 90° CW rotation across each row and a colour cycle down each column. Leave one cell blank and try to complete it from the rules.
2. Take a published Raven-style matrix and write the row rule and column rule in words before looking at the options. If you cannot state the rules in words, you have not understood the matrix.
3. Time yourself on 10 matrices. Aim for under 30 seconds each. If you exceed 30 seconds on a matrix, flag it for review and move on — do not let one item sink your section timing.

The final section moves from 2D figures to 3D spatial reasoning: folding, hidden faces, and paper folding.