Matrices

A matrix question gives you a grid of shapes (usually 3 rows and 3 columns) with one square missing. You need to find the shape that completes the grid. The key idea is that every row and every column follows a rule — and sometimes the diagonals do too.

Matrices are often considered one of the harder NVR question types, but once you learn the strategy, they become much more manageable.

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What Does a Matrix Look Like?

Imagine a 3×3 grid like a noughts-and-crosses board:

+---------+---------+---------+
| Shape 1 | Shape 2 | Shape 3 |
+---------+---------+---------+
| Shape 4 | Shape 5 | Shape 6 |
+---------+---------+---------+
| Shape 7 | Shape 8 |    ?    |
+---------+---------+---------+

The missing shape is usually in the bottom-right corner (but not always). You need to find the answer that correctly completes the pattern.

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The Golden Rule of Matrices

Each row follows the same rule(s), AND each column follows the same rule(s).

This means the answer must work when you read across the bottom row AND when you read down the right column. If your answer does not work for both, it is wrong.

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Common Matrix Patterns

Pattern 1: Each Row Contains the Same Set of Elements

Each row (and each column) contains one of each type.

Example:

Row 1: Circle,   Square,   Triangle
Row 2: Square,   Triangle, Circle
Row 3: Triangle, Circle,   ?

Each row contains one circle, one square, and one triangle. The missing shape in Row 3 must be a square (because Row 3 already has a triangle and a circle).

Check the column: Column 3 has Triangle, Circle, ? — it needs a Square. Confirmed!

Pattern 2: Progressive Change Across Rows

A property changes step by step across each row.

Example (shading):

Row 1: Black circle,    Grey circle,    White circle
Row 2: Black square,    Grey square,    White square
Row 3: Black triangle,  Grey triangle,  ?

The pattern: shading goes from Black → Grey → White in each row. The missing shape is a White triangle.

Pattern 3: Combination / Overlay

The third shape in each row is created by combining the first two shapes.

Example:

Row 1: Vertical line  |  Horizontal line  |  Plus sign (+)
Row 2: Forward slash  |  Backward slash   |  X shape
Row 3: Circle         |  Square           |  ?

The rule: shape 1 + shape 2 = shape 3 (the elements are overlaid). The missing shape should be a circle with a square overlaid on it (circle inside a square, or vice versa).

Pattern 4: Rotation Across Rows

An element rotates as you move across each row.

Example:

Row 1: Arrow UP,    Arrow RIGHT,  Arrow DOWN
Row 2: Arrow LEFT,  Arrow UP,     Arrow RIGHT
Row 3: Arrow DOWN,  Arrow LEFT,   ?

Each row shows the arrow rotating 90° clockwise. Row 3: DOWN → LEFT → UP.

Pattern 5: Increasing / Decreasing Count

The number of elements changes across the row or down the column.

Example:

Row 1: 1 dot,  2 dots,  3 dots
Row 2: 2 dots, 3 dots,  4 dots
Row 3: 3 dots, 4 dots,  ?

The missing value is 5 dots.

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Worked Example 1

+---------+---------+---------+
| Large   | Medium  | Small   |
| black   | black   | black   |
| circle  | circle  | circle  |
+---------+---------+---------+
| Large   | Medium  | Small   |
| grey    | grey    | grey    |
| circle  | circle  | circle  |
+---------+---------+---------+
| Large   | Medium  |    ?    |
| white   | white   |         |
| circle  | circle  |         |
+---------+---------+---------+

Across rows: Size goes Large → Medium → Small. Down columns: Shading goes Black → Grey → White.