Understanding the Matrix Pattern Question

The question asks us to carefully study a given pattern presented in a matrix format and determine the number that should replace the question mark (?). We need to find a logical rule or pattern that connects the numbers in the matrix.

The given matrix is:

We should look for relationships between the numbers, either across rows or down columns. Often, the number in the third row is derived from the numbers in the first two rows of the same column, or similarly for rows.

Analyzing the Pattern in the Columns

Let's examine the columns to see if there is a consistent pattern relating the first two numbers to the third number in each column.

Consider the first column:

• Top number: 9
• Middle number: 5
• Bottom number: 28

How can we get 28 from 9 and 5? Let's try some common operations:

• Sum: $9 + 5 = 14$. $14 \times 2 = 28$. This looks promising. Let's check if multiplying the sum by 2 works for other columns.
• Difference: $|9 - 5| = 4$. Not 28.
• Product: $9 \times 5 = 45$. Not 28.

The pattern 2 \times (Sum of top two numbers) seems like a possibility for the first column.

Now, let's check this potential pattern on the second column:

• Top number: 4
• Middle number: 8
• Bottom number: 24

Using the pattern 2 \times (Sum of top two numbers):

$2 \times (4 + 8) = 2 \times 12 = 24$.

This matches the bottom number in the second column. So, the pattern The number in the third row is twice the sum of the numbers in the first two rows of that column seems to hold for the first two columns.

Applying the Pattern to Find the Missing Number

We will now apply the identified pattern to the third column to find the missing number (represented by the question mark?).

In the third column:

• Top number: ?
• Middle number: 5
• Bottom number: 42

Let the missing number be $x$. According to the pattern, the number in the third row (42) is twice the sum of the numbers in the first two rows ($x$ and 5).

So, we can write the equation:

$$2 \times (x + 5) = 42$$

Now, we solve for $x$:

Divide both sides by 2:

$$x + 5 = \frac{42}{2}$$

$$x + 5 = 21$$

Subtract 5 from both sides:

$$x = 21 - 5$$

$$x = 16$$

Therefore, the missing number that replaces the question mark is 16.

Conclusion: The Missing Number

Based on the consistent pattern found in the first two columns, where the bottom number is twice the sum of the top two numbers, the missing number in the third column is 16.

Revision Table: Key Pattern Concepts

Additional Information: Types of Reasoning Patterns

Matrix reasoning questions like this one test your ability to identify logical patterns. These patterns can involve various relationships:

• Arithmetic Progressions: Numbers increase or decrease by a constant difference.
• Geometric Progressions: Numbers increase or decrease by a constant ratio (multiplication or division).
• Combinations of Operations: The pattern might involve adding, subtracting, multiplying, or dividing different elements in a specific sequence.
• Squaring or Cubing: Numbers might be related to the squares or cubes of other numbers.
• Sum/Difference/Product Relationships: One number might be the sum, difference, or product of other numbers.
• Positional Rules: The pattern might depend on the position of the numbers in the matrix (e.g., the number in the bottom right is the sum of the top left and top right).

Always start by examining rows and columns separately. Test simple patterns first before looking for more complex ones. Consistency across the known elements is key to confirming the pattern.