Matrices
Complete a grid by finding the rules for rows and columns.
What is Matrices?
Matrix questions show you a grid (usually 3×3) of shapes where one cell is missing. Each row and each column follows a rule. You need to work out the rules and find the shape that completes the grid.
These questions are like sequences in two dimensions – you need to track patterns going across AND going down. The missing piece is usually in the bottom-right corner.
Step-by-Step Method
Study the rows
Look at each complete row. What changes from left to right? What stays the same?
Study the columns
Look at each complete column. What changes from top to bottom? What stays the same?
Combine the rules
The missing shape must follow BOTH the row rule and the column rule. Use both to narrow down the answer.
Check your answer
Place your answer in the grid and verify it follows the pattern in its row and its column.
Worked Examples
A 3×3 grid has: Row 1: circle, square, triangle. Row 2: circle, square, triangle. Row 3: circle, square, ?. Each row has the same shapes. Each column has all the same shape. What fills the gap?
Working
- Row rule: each row contains circle, square, triangle.
- Column rule: column 1 is all circles, column 2 is all squares, column 3 is all triangles.
- The missing shape is in row 3, column 3.
- Row 3 needs a triangle. Column 3 needs a triangle. Both agree.
In a 3×3 grid, row 1 has shapes getting larger left to right. Row 2 has shapes getting larger left to right. Column 1 shows white shading, column 2 shows grey, column 3 shows black. The bottom-right cell is missing.
Working
- Row rule: shapes get larger left to right, so the missing shape is large.
- Column rule: column 3 shapes are black, so the missing shape is black.
- Answer: a large black shape (check what shape type the row uses).
Common Mistakes
Only checking the row pattern and ignoring the columns (or vice versa).
Always check BOTH rows and columns. The answer must satisfy both directions.
Getting confused when the rules are about combining or overlapping elements rather than simple progression.
Some matrices use overlay rules (e.g. shapes from cells 1 and 2 combine to make cell 3). Check for this if simple progression does not work.
Top Tips
- Start with the complete rows and columns – they give you the most information.
- Common rules: rotation, size change, shading change, number of elements, and shape overlay.
- If simple rules do not work, check if shapes combine (e.g. one shape placed on top of another).
- Draw the answer before checking the options – this prevents you being influenced by wrong answers.
Ready to practise?
Put these techniques into action with our free practice papers.
Practise Non-Verbal Reasoning Questions