9.2 Matrix Reasoning and Figural Analogies

9.2 Matrix Reasoning and Figural Analogies

Where a figure series runs in a single line, a matrix runs in two directions at once and a figural analogy compresses the whole idea into one transformation applied twice. Both appear in the non-verbal part of the AdUCET aptitude test. Adamson University publishes no content outline, so nobody can tell you how many of each to expect; what is worth knowing is that these two formats reward a very different discipline from the linear series covered in section 9.1. In a series you extrapolate. In a matrix and an analogy you identify a relation and then apply it somewhere else.

Reading a 3-by-3 matrix

Eight cells are filled and one is missing, almost always the bottom-right.

+-----------+-----------+-----------+
|   R1C1    |   R1C2    |   R1C3    |
+-----------+-----------+-----------+
|   R2C1    |   R2C2    |   R2C3    |
+-----------+-----------+-----------+
|   R3C1    |   R3C2    |   [  ?  ] |
+-----------+-----------+-----------+

Work in this order:

1. Read row 1 left to right. Most matrices are built row-wise, so this is where the rule usually is.
2. Confirm on row 2. A rule that works on one row is a guess. A rule that works on two rows is a rule.
3. If the rows give nothing, read down the columns.
4. If both fail, look for a whole-grid rule — every row and column carrying each variant once, or every row totalling a fixed number.
5. Check diagonals last. They are rare.

In a well-built matrix the row rule and the column rule are usually both satisfied, and that gives you a free check: predict the missing cell from row 3, then verify it from column 3. If the two predictions disagree, one of your two rules is wrong.

Boolean line combination

The commonest third-cell rule combines line segments. Suppose every cell is drawn on the same small lattice and may carry any of four strokes: a vertical bar, a horizontal bar, a forward slash and a back slash. Take one pair of cells and apply each operation in turn.

  Cell 1 = { vertical , horizontal }
  Cell 2 = { horizontal , forward slash }

  Union (OR)         ->  { vertical , horizontal , forward slash }   3 strokes
  Intersection (AND) ->  { horizontal }                              1 stroke
  Exclusive OR (XOR) ->  { vertical , forward slash }                2 strokes

Deciding which operation is running takes one count. If cell 3 carries more strokes than either input, it is a union. If it carries fewer than both, it is an intersection. If it carries strokes drawn from both inputs but is missing whatever they had in common, it is XOR. Exclusive OR is the examiners' favourite precisely because the shared stroke vanishing looks like a printing error to an untrained eye.

Addition, subtraction and constant totals

The same three ideas appear with countable objects instead of strokes.

Subtraction across a row. The third cell holds the first cell's elements minus the second cell's.

Row 1 gives $6 - 2 = 4$ and row 2 gives $8 - 3 = 5$, so the rule is confirmed twice; row 3 gives $7 - 5 = 2$. Addition works identically with the third cell holding everything from the first two.

Constant sum. Every row totals the same number. If row 1 is $4 + 5 + 3 = 12$ and row 2 is $7 + 2 + 3 = 12$, then row 3 showing 6 and 1 needs $12 - 6 - 1 = 5$ in the missing cell.

Constant count. A gentler variant: every row uses each of three fixed counts — say 1, 3 and 5 — exactly once in some order. The missing cell is then whichever count row 3 has not yet spent.

Distribution of three

Here each of three variants appears exactly once in every row and exactly once in every column, a Latin square.

            Column 1           Column 2          Column 3
 Row 1      white circle       hatched square    black triangle
 Row 2      black square       white triangle    hatched circle
 Row 3      hatched triangle   black circle      [    ?    ]

Solve the two attributes separately, never together. Row 3 has already used the triangle and the circle, so the missing shape is a square. Row 3 has already used hatched and black, so the missing fill is white. Now cross-check down column 3: its shapes so far are triangle and circle, which again forces a square, and its fills so far are black and hatched, which again forces white. The answer is a white square, confirmed twice. Two independent distributions running at once — one for shape, one for fill — is the standard difficulty upgrade on this format.

Two-rule matrices

One rule may run across the rows while a different rule runs down the columns.

            Column 1        Column 2        Column 3
 Row 1      triangle, 2     triangle, 4     triangle, 6
 Row 2      square, 3       square, 5       square, 7
 Row 3      circle, 1       circle, 3       [   ?   ]

The count rule runs across the rows at $+2$ per cell; the shape rule runs down the columns, reading triangle, square, circle from top to bottom. The missing cell is a circle containing 5 dots, and column 3 confirms the shape independently. Notice that neither rule alone would have solved the cell — the count rule fixes the 5 and the shape rule fixes the circle.

Figural analogies

An analogy is written A : B :: C : ? and means: whatever was done to A to produce B, do exactly that to C.

1. Describe A to B in one sentence, and say what did not change. "The hexagon became a triangle; the interior dot stayed put and the fill stayed white."
2. Check the description is the only one that fits. A single turned arrow can equally be a flip; look for an asymmetric feature to settle it. The decisive test lives in section 9.3.
3. Apply the transformation to C literally, feature by feature.
4. Predict, then read the options.

Worked analogy 1. A white hexagon with a single dot at its centre is to a white triangle with a single dot at its centre, as a white octagon with a single dot at its centre is to what? The transformation halves the side count and changes nothing else, since $6 \to 3$. Applying it, $8 \to 4$, so the answer is a white square with a single dot at its centre. The rival description "remove three sides" also fits $6 \to 3$, and it would give a pentagon; you cannot separate the two from one pair, so read the options and see which candidate rule is actually offered — the halving reading is the one examiners intend when the numbers permit it.

Worked analogy 2. A small white square is to a large white square containing a small black circle at its centre, as a small white pentagon is to what? Two things happened at once: the figure grew, and a black centre dot appeared. Apply both to the pentagon and the answer is a large white pentagon with a black dot at its centre. Both half-answers will be on the list.

Common traps

• The style-match trap. An option drawn in C's own style — same shape, same size, same fill — with the transformation simply not applied. It is C itself, or C with one cosmetic change, and it looks right because it echoes the term you just read.
• Copying B. An option that reproduces B, or B with a trivial change. It satisfies the eye's demand for a match with the second term.
• Partial transformation. Where two things changed from A to B, only one is applied to C.
• A rule fitted to one row. In matrices this is the main source of wrong answers. Confirm on two rows.
• Ignoring the column check. When two options survive the row rule, the column rule almost always separates them.