Figure series, mirrors, embedded shapes and paper folding.
About Figure Matrix
A figure matrix is a three by three grid of figures with the last cell missing. Each row is built from its own first figure by the same two changes, so the missing figure is the first figure of the last row with those changes applied.
What you need to understand
The rule runs along the rows: the second figure of a row is the first changed one way, and the third is the first changed another way.
The rule is the same in every row, but each row starts from a different figure.
The changes are reflections and turns, not changes of size.
A mirror reflection swaps left and right, a water reflection swaps top and bottom, and a half turn swaps both.
The third column is not the second column changed again. It is the first column changed a second way, which is a different thing.
Two complete rows are given, which is what pins the rule down. One row alone could be read in more than one way.
The missing cell is in the last row, so the figure to change is the first figure of that row.
How to work through these questions
Compare the first and second figures of the top row and note the change, feature by feature.
Compare the first and second figures of the middle row. The change there must be the same, and it confirms the reading.
Now compare the first and third figures of the top row, and note that change in the same way.
Confirm it against the middle row as well.
Take the first figure of the last row and apply the change that the third column shows.
Check the answer against the other cells of the last row to be sure nothing else is left to change.
Compare with the options and choose the one that matches exactly.
Mistakes that cost marks
Reading down the columns when the rule runs along the rows.
Treating the third column as the second column changed again, which gives a figure changed twice over.
Applying only the change in the second column and forgetting the third.
Assuming the same starting figure for every row, when each row begins from a different one.
Choosing a figure from another row that looks similar.
Reading a mirror reflection as a water reflection, so that one axis comes out right and the other wrong.
Worked example
In a three by three matrix, the first row holds a figure, then its mirror image, then its water image. The third row holds a different figure, then its mirror image, and the last cell is missing. Which figure completes the matrix?
Compare the first and second cells of the first row. Left and right have changed places, so the second column holds the mirror image of the first.
Compare the first and third cells of the first row. Top and bottom have changed places, so the third column holds the water image of the first.
Check both readings against the second row, where the changes must be the same.
The missing cell is in the third column of the last row, so it must hold the water image of that row first figure.
Find the option that is the first figure of the last row reflected in water.
Answer: The water image of the first figure of the last row.
Practice questions with answers
A few Figure Matrix questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
In this matrix each row is built from its own first figure in the same two ways. The last cell is missing.
Which figure completes the matrix?
A
B
C
D
Answer: Option D — with explanation
Compare the first column with the second: in every row the figure is reflecting it in water lying below it, which swaps top and bottom.
Compare the first column with the third: in every row the figure is turning it through half a turn, which swaps both.
Applying the change shown in the third column to the first figure of the last row gives the answer.
Common mistakes
- the third figure reflecting it in water lying below it, which swaps top and bottom, when the third column of the matrix shows the other change
- the third figure reflecting it in a mirror standing on its right, which swaps left and right, when the third column of the matrix shows the other change
- the third figure leaving the figure exactly as it was, when the third column of the matrix shows the other change
Question 2
In this matrix each row is built from its own first figure in the same two ways. The last cell is missing.
Which figure completes the matrix?
A
B
C
D
Answer: Option D — with explanation
Compare the first column with the second: in every row the figure is reflecting it in a mirror standing on its right, which swaps left and right.
Compare the first column with the third: in every row the figure is reflecting it in water lying below it, which swaps top and bottom.
Applying the change shown in the third column to the first figure of the last row gives the answer.
Common mistakes
- the third figure leaving the figure exactly as it was, when the third column of the matrix shows the other change
- the third figure reflecting it in a mirror standing on its right, which swaps left and right, when the third column of the matrix shows the other change
- the third figure turning it through half a turn, which swaps both, when the third column of the matrix shows the other change
Question 3
In this matrix each row is built from its own first figure in the same two ways. The last cell is missing.
Which figure completes the matrix?
A
B
C
D
Answer: Option B — with explanation
Compare the first column with the second: in every row the figure is reflecting it in a mirror standing on its right, which swaps left and right.
Compare the first column with the third: in every row the figure is turning it through half a turn, which swaps both.
Applying the change shown in the third column to the first figure of the last row gives the answer.
Common mistakes
- the third figure reflecting it in a mirror standing on its right, which swaps left and right, when the third column of the matrix shows the other change
- the third figure leaving the figure exactly as it was, when the third column of the matrix shows the other change
- the third figure reflecting it in water lying below it, which swaps top and bottom, when the third column of the matrix shows the other change
Question 4
In this matrix each row is built from its own first figure in the same two ways. The last cell is missing.
Which figure completes the matrix?
A
B
C
D
Answer: Option C — with explanation
Compare the first column with the second: in every row the figure is turning it through half a turn, which swaps both.
Compare the first column with the third: in every row the figure is reflecting it in a mirror standing on its right, which swaps left and right.
Applying the change shown in the third column to the first figure of the last row gives the answer.
Common mistakes
- the third figure turning it through half a turn, which swaps both, when the third column of the matrix shows the other change
- the third figure leaving the figure exactly as it was, when the third column of the matrix shows the other change
- the third figure reflecting it in water lying below it, which swaps top and bottom, when the third column of the matrix shows the other change
Question 5
In this matrix each row is built from its own first figure in the same two ways. The last cell is missing.
Which figure completes the matrix?
A
B
C
D
Answer: Option A — with explanation
Compare the first column with the second: in every row the figure is turning it through half a turn, which swaps both.
Compare the first column with the third: in every row the figure is reflecting it in a mirror standing on its right, which swaps left and right.
Applying the change shown in the third column to the first figure of the last row gives the answer.
Common mistakes
- the third figure leaving the figure exactly as it was, when the third column of the matrix shows the other change
- the third figure turning it through half a turn, which swaps both, when the third column of the matrix shows the other change
- the third figure reflecting it in water lying below it, which swaps top and bottom, when the third column of the matrix shows the other change
Frequently asked questions
Does the rule run along the rows or down the columns?
Along the rows in this kind of question. Each row changes its own first figure in the same two ways that every other row uses.
Why do I need two complete rows instead of one?
Because one row on its own can often be read in more than one way. A second row that obeys the same rule is what shows which reading is intended.
Is the third figure the second figure changed again?
No. The third figure is the first figure changed a second way. Treating it as the second figure changed again would apply the change twice.
What kinds of change appear in a matrix?
Reflections and turns. A mirror reflection swaps left and right, a water reflection swaps top and bottom, and a half turn swaps both. Size never changes.
Which figure do I apply the rule to?
The first figure of the incomplete row. The other changes in that row show what has already been done to it.
Take the Figure Matrix test
Two timed papers on the same syllabus — sit the foundation paper first,
then the advanced one. Both use the real exam paper format with a full
step-by-step review of every question once you submit.
