11+ Matrix Reasoning: How to Solve Every Matrix Question

Matrix reasoning questions put shapes into a grid, usually 2x2 or 3x3, where each shape changes according to a hidden rule as you move across or down, and the child has to work out which shape completes the missing square. There's no vocabulary and no arithmetic here; it's pure pattern-spotting, which makes it either very fast marks or a frustrating guessing game, depending on whether the child has a systematic method.

What this question type covers

• 2x2 matrices: four squares, three shapes shown, one missing; find the rule linking them
• 3x3 matrices: nine squares, eight shapes shown, one missing; usually involves two rules working together (one across each row, one down each column)
• Rules typically involve: rotation, reflection, changes in size, changes in shading or colour, addition or removal of an element, or a count that increases or decreases

Worked example 1: 2x2 matrix, single rule

Grid description: top-left shows a small triangle; top-right shows a medium triangle; bottom-left shows a large triangle; bottom-right is blank.

Check across the top row first, then down the left column, to see which direction the rule operates in.

• Across the top row: small triangle to medium triangle (the shape grows)
• Down the left column: small triangle to large triangle (the shape also grows, more steeply)

The rule is "size increases," consistently in both directions. Following the pattern to the bottom-right square, the shape should be a triangle even larger than the bottom-left one.

Answer: an extra-large triangle.

Worked example 1b: 2x2 matrix, two rules combined

Grid description: top-left shows a small white circle; top-right shows a small black circle; bottom-left shows a large white circle; bottom-right is blank.

Check whether the row and column rules are different from each other.

• Across the top row: white circle to black circle (shading changes)
• Down the left column: small circle to large circle (size changes)

Two separate rules are operating: one across, one down. The missing square needs both applied together: large and black.

Answer: a large black circle.

This is a harder variant because a child who only checks one direction will miss half the rule and pick an answer that's only half-correct, for example choosing a large white circle (right size, wrong shading) or a small black circle (right shading, wrong size).

Worked example 2: 3x3 matrix

Grid description: each row shows a shape rotating 90 degrees clockwise as you move left to right; each column shows the shape's shading getting one shade darker as you move down; bottom-right square is missing.

Check rows and columns separately, exactly as in worked example 1b, then combine both rules for the missing square.

• Row rule: rotate 90 degrees clockwise moving left to right
• Column rule: shading gets darker moving top to bottom

The missing square needs the shape rotated to match its row position and shaded to match its column position. Answer: the shape rotated to the row's expected angle, in the darkest shading level used in that column.

3x3 matrices are usually two 2x2-style rules layered together, which is why mastering the two-rule habit in worked example 1b transfers directly.

The mistakes we see most often

1. Checking only the row or only the column, missing a second rule operating in the other direction. This is the single most common error, especially on 3x3 grids.
2. Assuming a rule seen in one row is the whole rule, without checking whether a second, independent rule is layered on top.
3. Getting distracted by shape identity instead of the transformation. The specific shape (triangle, circle, square) is rarely what's being tested; it's what happens to it (rotation, size, shading, count) that matters.
4. Guessing from the answer options first rather than working out what the missing square should look like independently, then matching it to an option.
5. Losing track of direction of rotation or reflection. Clockwise vs anticlockwise, or a mirror flip vs a rotation, are easy to confuse under time pressure without a consistent checking habit.

A practice strategy parents can use at home

Before timing any practice, have your child describe the rule out loud in words ("it gets bigger and darker each time") before picking an answer. Verbalising the rule forces them to check both directions of a grid rather than jumping to the first shape that looks about right, and it's an easy habit to check as a parent even without expert knowledge of NVR yourself. You're listening for a clear, complete rule, not judging the answer.