Matrices & Completing the Grid

<p>Matrix questions show a 3×3 grid (or sometimes 2×2 or 4×4) of shapes with one cell empty. You must work out the pattern across rows and columns to find the missing shape. These questions combine pattern-spotting with logical thinking.</p>

<h2>What Is a Matrix?</h2>

<p>A matrix is a grid where each row follows a rule and each column follows a rule. The rules might involve the shape, size, shading, number of elements, orientation, or position of objects within each cell.</p>

<h2>How to Solve a 3×3 Matrix</h2>

<ol>

<li><strong>Look across each row:</strong> What changes from left to right? What stays the same?</li>

<li><strong>Look down each column:</strong> What changes from top to bottom? What stays the same?</li>

<li><strong>Identify the rules:</strong> There is usually one rule for rows and one for columns</li>

<li><strong>Apply both rules</strong> to the empty cell to determine the missing shape</li>

<li><strong>Verify:</strong> Check that your answer works for both the row AND the column it belongs to</li>

</ol>

<h2>Worked Example 1: Shape Changes</h2>

<p>Imagine a 3×3 grid:</p>

<p>Row 1: circle, triangle, square<br>

Row 2: circle, triangle, square<br>

Row 3: circle, triangle, ?</p>

<p>Each row contains a circle, triangle, and square. The missing shape is a square.</p>

<p>Check column 3: square, square, square — yes, column 3 is all squares. ✓</p>

<h2>Worked Example 2: Two Features</h2>

<p>Row 1: small black circle, medium black circle, large black circle<br>

Row 2: small grey circle, medium grey circle, large grey circle<br>

Row 3: small white circle, medium white circle, ?</p>

<p>Row rule: size increases (small → medium → large). Column rule: shading changes (black → grey → white).<br>

Missing shape: large white circle. ✓</p>

<h2>Worked Example 3: Elements Change</h2>

<p>Row 1: 1 dot, 2 dots, 3 dots<br>

Row 2: 2 dots, 3 dots, 1 dot<br>

Row 3: 3 dots, 1 dot, ?</p>

<p>Each row contains 1, 2, and 3 dots (in different orders). Row 3 has 3 and 1 — it needs 2 dots.<br>

Check columns: Column 3 has 3, 1, ? — it needs 2 dots. ✓</p>

<h2>Common Matrix Rules</h2>

<ul>

<li><strong>Each row/column contains one of each type</strong> (like a Sudoku for shapes)</li>

<li><strong>Progressive change:</strong> something increases or decreases across the row/column</li>

<li><strong>Rotation:</strong> a shape rotates by the same amount in each cell</li>

<li><strong>Combination:</strong> the third cell combines elements from the first two cells</li>

<li><strong>Subtraction:</strong> the third cell shows what is different between the first two</li>

</ul>

<h2>The Combination Rule</h2>

<p>In some matrices, the third cell in each row contains the shapes from the first two cells combined (overlaid on top of each other). Or, conversely, the third cell shows only the parts that are NOT shared between the first two.</p>

<p><strong>Example:</strong> Cell 1 has a horizontal line. Cell 2 has a vertical line. Cell 3 has a cross (both lines combined).</p>

<h2>Harder Matrices</h2>

<p>In more difficult questions, multiple features change simultaneously. One feature follows the row rule and another follows the column rule. Break the problem into separate features and solve each one independently.</p>

<p><strong>Example:</strong> Rows determine the outer shape (circle, triangle, square across each row). Columns determine the inner shape (dot, cross, star down each column). The missing cell must have the outer shape that completes the row AND the inner shape that completes the column.</p>

<h2>Practice Tips</h2>

<ul>

<li>Always check BOTH the row and the column — your answer must satisfy both rules</li>

<li>If you cannot see the row pattern, try the columns first — sometimes one direction is easier to spot</li>

<li>For combination-type rules, sketch the first two cells on top of each other and see if it matches the third</li>

<li>Practise with Sudoku puzzles — they use similar logic (each row and column must contain one of each)</li>

<li>Use elimination: if you have 5 answer options, check each against the row and column rules and cross off any that fail</li>

</ul>