Matrices
A grid of pictures with one box left empty. Here's how to find the rule along the rows and the rule down the columns, and pick the picture that fits both.
Chapter 1
What are matrix questions?
A matrix is a grid of boxes, 2 × 2 or 3 × 3, like a noughts and crosses board. One box is empty, and you pick the picture that goes in it.
Each picture is made of parts: a shape, its shading, its size, its outline, dots, a smaller shape inside. Each part follows a rule along the rows, down the columns, or both. The right picture fits its row and its column.
The rows. Read across, left to right.
The columns. Read down, top to bottom.
Which goes in the empty box?
An NVR question from Rush, level 2.
Four rules grids use
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It stays the same
A part can stay the same all along a row or all down a column. Here the shape stays the same along each row and the shading down each column, so the gap is a black square.
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It turns
Along each row the arrow turns a quarter clockwise, and down each column it turns black. So the gap is a black arrow pointing right.
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One of each
Like a sudoku: every row and column has one circle, one square and one triangle. The bottom row has no square yet, so that’s the gap. Check: there are only two squares.
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It grows along the row
Small, medium, big along every row, and each row starts again with its own shape. So the gap is a big circle. Trap: a medium one, copied from the box beside the gap.
Chapter 2
Solve it in 3 steps.
Here’s the question, one part at a time.
The question
Which goes in the empty box?
An NVR question from Rush, level 2.
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Step 1
Take one part: the shading.
Ignore the dots for now, and the dashed ring, which never changes. Every row has one black, one white and one striped diamond, and so does every column. The bottom row has striped and black, so the gap is white. The right column says white too.
Its row and its column both need white.
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Step 2
Then the next part: the dots.
Now only the dots. Along every row they go 0, 1, 2, and down each column they stay the same. The gap is in the right column, so it has 2 dots.
012The same number all the way down.
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Step 3
Build the box, then cross out.
The gap is a white diamond with 2 dots. Options 2 and 3 aren’t white: out. Options 1 and 4 are white, but with 0 and 1 dots: out. That leaves option 5.
12345
Three mistakes this grid sets
| The mistake | Do this instead |
|---|---|
| Picking a box you’ve already seen. Every wrong option here is copied from the grid. Option 2 is the top left box again. | Build the gap from the rules first, then look for it. |
| Stopping at one rule. Option 1 is white, as its row and column need, but it has no dots. | Check every option against every rule you found. |
| Counting from the wrong column. Option 4 is white with 1 dot. That’s the middle column’s number, not the gap’s. | Read the dots down the gap’s own column. |
So it’s option 5. It’s white, which its row and its column both need, with 2 dots, like the rest of the right‑hand column.
Remember: Row rule. Column rule. Fit both.
Chapter 3
The catches.
Rush asks these four kinds too. Each card gives the rule, an example and the trap.
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Mirror grids: one half turn
Turn the top left box half way round.
Each box is a mirror image of the box beside it and the box above it. Flipping across and then down is the same as one half turn. Trap: a copy of the box beside the gap or above it, not flipped.
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toprightbottom
Symbols that move round
Follow one symbol along the row.
Every symbol moves one place clockwise. The circle goes top, then right, so next it’s at the bottom. Then check one more. Trap: the row’s first box again, a turn too far, or two symbols swapped.
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Small changes count
Check every part, even the small ones.
The change can be small: the shape inside, or a dashed ring. Here the shape inside changes along each row, and the ring appears down each column. Trap: the right shape inside, but no ring.
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Round grids
Look straight across the middle first.
Some grids are a ring or a hexagon. Opposite boxes often match, so the gap copies the one across from it. If not, try every other box. Trap: copying a box next to the gap.
Chapter 4: In the exam
The gap can be anywhere.
In the exam the empty box isn’t always bottom right, and the options often hide a mirror image or a turn the wrong way. Rush’s square grids always leave the bottom right box empty, but the same steps work.
Which goes in the empty box?
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Step 1
Find a full row and a full column.
The gap is top right. The bottom row is full, so it shows the rule across. The left column is full, so it shows the rule down.
The bottom row and the left column are full.
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Step 2
The rule across: it turns.
Along the bottom row the L turns a quarter clockwise. So the gap is the top left L turned a quarter clockwise. B has turned the other way and E hasn’t turned at all: both out.
ABCDE -
Step 3
The rule down: it turns black.
Down the left column, white becomes black. The gap is in the top row, so it stays white. D is black: out.
ABCDE -
Step 4
Turned, or flipped?
A and C are both white and on their side. Turn the top left L a quarter clockwise and its foot hangs down at the left, as in C. In A the foot sticks up: it’s a mirror image, and no turn makes one. Out.
ABCDE
So it’s C. It’s the top left L turned a quarter clockwise, still white, and not flipped over.
Remember: Full row. Full column. Then the mirror.
Other ways the exam asks it
- Rules on the diagonal. If the rows and columns show no rule, read corner to corner. Each diagonal can hold the same shape.
- Two boxes make a third. One box in a row can be the other two drawn on top of each other.
- Three or four rules at once. Rush keeps to two. The papers sometimes use more, and one part at a time still cracks them.
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Take one part: the shading.
- Then the next part: the dots.
- Build the box, then cross out.
Remember: Row rule. Column rule. Fit both.
For parents.
What parents ask us about matrices.
Questions parents ask
What are matrices in non-verbal reasoning?
A matrix is a grid of figures that follow rules along the rows and down the columns, with one box missing. You work out the rules and choose the figure that completes the grid. Some books call them grids or grid completion.
Should you read along the rows or down the columns first?
Either, but do both. Many grids use one rule across and a different rule down. If you stop after the first rule you'll usually be left with two or three options that fit. If neither direction shows a rule, read the diagonals: in some paper grids the boxes running corner to corner match.
What are star matrices?
They're grids that aren't square: a star, a hexagon or a ring of boxes with one left empty. The rules link boxes opposite each other or next to each other, so the gap usually matches the box straight across from it, or every other box round the ring. Rush asks the hexagon and ring kinds; the star kind is found in some practice papers.
Are 2 × 2 grids easier than 3 × 3?
They have fewer boxes, but you only see each pattern once, so there's less to go on. A 3 × 3 grid gives you two full rows to confirm a rule before you use it on the third.
Rush: 11+ Practice & KS2 SATsPractise matrices in Rush.
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