9.3 Matrix Completion

Rows and columns are two clocks, not one pretty picture

A matrix-completion item shows a grid of figures with one cell missing, usually the bottom-right cell of a 3×3 teaching grid. Your job is to choose the figure that belongs in the gap. The skill is still analytical aptitude inside QFD Core Abilities, not Mechanical Reasoning. The grid is not a station-bay layout and not a circuit. It is a set of visual rules that run across and down.

QFD does not publish whether live Core Abilities items use 3×3 matrices, 2×2 matrices, or another layout, and it does not publish how many such items appear. This section uses a labelled 3×3 because that is the clearest way to teach row and column discipline. Treat the labels as study scaffolding, not as a claimed official format.

Call the cells by coordinates so you can talk without a picture:

• Row 1: A11, A12, A13
• Row 2: A21, A22, A23
• Row 3: A31, A32, A33 (missing)

A11 is top-left. A13 is top-right. A31 is bottom-left. A33 is the gap.

The governing idea: the missing cell must satisfy the row-3 rule and the column-3 rule. If you only complete the row, you will pick a distractor that the column rejects. If you only complete the column, you will pick a distractor that the row rejects.

Worked matrix — shape across, fill down

Build this teaching matrix in words:

• A11: empty triangle. A12: empty square. A13: empty circle.
• A21: hatched triangle. A22: hatched square. A23: hatched circle.
• A31: filled triangle. A32: filled square. A33: ?

Row rule (left to right): shape cycles triangle, square, circle. Every row uses each shape once, in that order.

Column rule (top to bottom): fill cycles empty, hatched, filled. Every column uses each fill once, in that order.

Row 3 already has filled triangle and filled square, so the row wants a circle. Column 3 already has empty circle and hatched circle, so the column wants filled. Both clocks name the same cell: filled circle. That double naming is how you know you are finished.

Now the traps, described as figures rather than as letters:

• An empty circle would finish a naive shape story for row 3 if you forgot that row 3 is already in the filled band.
• A filled square repeats a shape already used in row 3 and ignores the circle demand.
• A hatched circle would continue column 3's shape (circle) but would reuse hatched, which column 3 already spent in A23.
• A filled triangle repeats row 3's first shape.

Only filled circle is legal on both axes.

Worked matrix — count across, extra mark down

A second teaching grid uses numbers of marks, still abstract:

• Each row shows 1 dot, then 2 dots, then 3 dots from left to right.
• Each column adds a small tick count of 0, then 1, then 2 from top to bottom, drawn as a mark on the top edge of the cell.

So A11 has 1 dot and 0 ticks. A13 has 3 dots and 0 ticks. A31 has 1 dot and 2 ticks. A32 has 2 dots and 2 ticks. A33 must have 3 dots (row clock) and 2 ticks (column clock). An option with 3 dots and 1 tick completes the row's count and undercounts the column ticks. An option with 2 dots and 2 ticks completes the column ticks and breaks the row's rise to 3.

This is the stacked-rule lesson from Section 9.1, now laid on two axes instead of along a line. One clock is horizontal. One clock is vertical. The gap is their intersection.

Worked matrix — distribution (one of each)

Some matrices do not cycle in a fixed left-to-right order. They distribute a set of marks so that each row and each column contains one of each family, like a tiny Latin square.

Teaching set of families: star, moon, sun.

• Row 1: star, moon, sun.
• Row 2: moon, sun, star.
• Row 3: sun, star, ?

Row 3 already has sun and star, so it still needs moon. Column 3 already has sun (A13) and star (A23), so it still needs moon. Both axes agree again: moon.

If row 3 had been sun, moon, ?, the row would demand star. Then you would still check column 3. If column 3 already contained a star, the grid would be inconsistent with a simple one-of-each rule, which means you misread a cell. Recount before you invent a second stacked clock. Distribution matrices are usually clean; messy leftover rules often mean you swapped moon and sun in A22.

How to read a matrix under OnVUE constraints

You cannot pencil a mini-table. Use this silent order:

1. Ignore the options until both axis phrases exist. Options exist to sell single-axis answers.
2. Read row 1 and row 2 completely. Name the horizontal clock (shape cycle, fill cycle, count, or one-of-each).
3. Read column 1 and column 2 completely. Name the vertical clock. It is allowed to be a different feature from the row clock. In the first worked grid, rows carried shape and columns carried fill.
4. Apply row 3's clock to the gap, producing a partial answer (circle, or 3 dots, or moon).
5. Apply column 3's clock to that partial answer. Keep the candidate only if both phrases survive.
6. Only then look at the options. Delete any figure that fails either phrase.

Traps on matrix completion

• Completing the row and never inspecting the column (or the reverse).
• Assuming the column uses the same feature as the row. Often it does not: shape across, fill down.
• Treating the grid as a sequence that snakes row by row like text. Matrix rules are axis rules, not a snake.
• Copying A22 into A33 because the centre looks like a theme. The centre is one cell, not a master key, unless you have proved it.
• Importing Mechanical Reasoning (the arrows would mesh like gears).
• Over-fitting a unique story for every cell instead of one row clock plus one column clock.

If row clocks and column clocks seem to demand two different figures, you have misread a given cell. Go back to A11 and re-name features. A well-built teaching matrix — and a well-built live item — has one legal intersection. QFD has not published worked official matrices; use that intersection test on independent practice, not as a claim about Pearson's item-writing software.

The pack says Core Abilities also measures capacity to learn new information. A matrix is exactly that: the rule is not in a firefighter manual. It is in the eight given cells. Learn it in 20 seconds, apply it once, move on. The pack also says you are not expected to get every item right. If a stacked matrix is still opaque after a row pass and a column pass, flag it and spend the time on a one-rule sequence you can finish.

Cover A33 with your thumb in imagination and re-derive it. Row 3 is already in the filled band and has used triangle and square, so it needs a circle. Column 3 is already in the circle family and has used empty and hatched, so it needs filled. The gap is the intersection: filled circle. Any option that is a circle but not filled, or filled but not a circle, failed one axis.