How to solve matrix reasoning: the rules behind every Raven matrix
Figure matrices are the most widespread item type in intelligence tests. John Raven published the Progressive Matrices in 1938, and the format has been part of practically every culture-fair test since, because it needs no language and no prior knowledge. The advantage for preparation: behind the apparent variety sits a small, fixed set of rules. Once you know them, you solve matrices not by looking but by working through a procedure.
Structure
A matrix consists of nine cells in three rows and three columns. The bottom-right cell is missing. Each cell shows a figure described by several attributes. The usual attributes are:
- Shape (circle, square, triangle, star, cross, arrow)
- Count of elements in the cell (one to five)
- Fill (filled or outline only)
- Rotation (in steps of 45 or 90 degrees)
- Less often: size, position within the cell, line weight
The key insight: each attribute follows its own rule, independently of the others. A matrix with four active attributes is not one hard item, it is four easy ones.
The four rule types
For each attribute there are essentially four ways it can behave across the grid.
1. Constant per row
All three cells in a row share the same value, the next row has a different one. Example: row one only circles, row two only triangles, row three only stars. The missing cell then has the shape of its row.
2. Constant per column
The same, vertically. The missing cell's shape is that of its column.
3. Progression
The attribute increases left to right or top to bottom, or both at once. For count: one, two, three elements per column. For rotation: 0, 45, 90 degrees. When row and column both count, the bottom-right cell has the highest value.
4. Distribution, the Latin square
Each value appears exactly once in every row and every column, as in Sudoku. Three shapes, and each row and column contains each of them once. The missing cell gets the value still missing from its row and its column. This rule is the most common in hard items and the most often overlooked, because the grid looks irregular at first glance.
For fill there is also the checkerboard: filled and empty alternate diagonally. Mathematically that is a progression modulo two, in practice you spot it immediately.
The elimination method
The procedure is the same for every matrix:
- Pick one attribute, preferably the most conspicuous, usually shape.
- Look at the first row. If all cells match, it is rule one. If every value differs, look at the first column: all the same means rule two, all different means rule four.
- Check the rule against the second row. If it holds, it is the rule.
- Derive which value the missing cell must have and cross out every answer option with a different one.
- Repeat with the next attribute until one option remains.
After two attributes one or two options usually remain. After three it is unambiguous. The advantage over trying to see the answer directly: the method works even when the matrix makes no sense at first glance, and it does not get slower with difficulty, only one attribute longer.
Typical traps
- Distractors with one wrong attribute. The wrong options usually differ from the right one in exactly one attribute. Anyone who checks only two of four attributes has a real chance of falling for one. Check every attribute, even when an option already looks plausible.
- Rotation of symmetric shapes. A circle cannot visibly rotate, a square only by 45 degrees. When rotation matters, the test uses asymmetric shapes. Conversely: if you only see circles and squares, rotation is not an active rule.
- Count confused with size. Three small circles and one large circle are different attributes. Count first, then compare size.
- Row and column at once. For progressions, check whether only the row, only the column, or both count. The top-left cell tells you: if it is the starting value for both directions, both count.
How long to practice
Matrices have the largest practice effect of all item types because the rules are so few. Anyone who knows the four rule types and the elimination method and has worked through two to three hundred matrices with explanations recognises the active rules within seconds. That is the state you want to reach before a test: the item is routine, and the time goes into the hard cases. For the wider picture see aptitude test preparation.
Frequently asked questions
What is the fastest way to solve matrix reasoning items?
Attribute by attribute. For shape, count, fill and rotation, determine the rule (constant per row, constant per column, each value once, or increasing) and after each rule cross out the options that violate it.
What are Raven's matrices?
An item type introduced by John Raven in 1938: a 3×3 grid of figures that follows rules across rows and columns, with the bottom-right cell missing. It is the most widespread item type in culture-fair intelligence tests.
Can you practice matrix reasoning?
Yes, and the practice effect is especially large for matrices because the number of possible rules is small. After two to three hundred items with explanations you recognise the active rules in seconds.
IQ Prep drills exactly these item types under test conditions, with an explanation after every answer. Free to try, full version $4.99 one time, no subscription.
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