In the following question, a matrix of numbers is given. These numbers follow a certain pattern either row wise or column wise. Study the pattern carefully and then choose the correct answer for the missing number. The matrix is: 72 24 6; 96 16 12; 108 ? 18. What number should replace the question mark ?

Introduction / Context:
This is a missing number puzzle based on a 3 by 3 numerical matrix. Each row appears to follow a specific mathematical relationship among its three entries. Our job is to identify this relationship from the known rows and apply it to find the missing number in the third row. Such problems assess numerical pattern recognition and the ability to test and confirm a suspected rule.

Given Data / Assumptions:

• The matrix has three rows: (72, 24, 6), (96, 16, 12), and (108, ?, 18).
• One number in the third row, middle column, is missing and denoted by a question mark.
• The pattern is consistent across the rows.
• We assume simple arithmetic operations such as addition, subtraction, multiplication, division, or combinations of these.

Concept / Approach:
To find the hidden pattern, we compare the structure of the first two rows where all entries are known. A useful idea is to see how the first, second, and third elements of each row relate to each other. For example, we check if the first number can be expressed in terms of the second and third numbers using multiplication or division. Once a relationship is identified that works for the first two rows, we apply it to the third row to find the missing middle element.

Step-by-Step Solution:
Step 1: Observe the first row: 72, 24, 6. If we multiply the second and third entries we get 24 * 6 = 144. Step 2: Notice that 72 is exactly half of 144. So, for the first row, first entry = (second entry * third entry) / 2. Step 3: Check the second row with this rule. The second row is 96, 16, 12. Multiply the second and third entries: 16 * 12 = 192. Step 4: Half of 192 is 192 / 2 = 96, which matches the first entry of the second row. So the same rule holds for the second row. Step 5: Apply this rule to the third row. The first entry is 108 and the third entry is 18. Let the missing middle entry be x. Then, according to the rule, 108 = (x * 18) / 2. Step 6: Rearranging gives x * 18 = 108 * 2 = 216, so x = 216 / 18 = 12. Step 7: Therefore, the missing number is 12.

Verification / Alternative check:
We can verify by reapplying the pattern. For the completed third row 108, 12, 18, the product of the second and third is 12 * 18 = 216. Half of 216 is 108, which matches the first number in the third row. Since the same relationship now holds for all three rows, the pattern is consistent and the solution is reliable.

Why Other Options Are Wrong:
If we substitute 16, 18, or 20 for the missing value, the relationship first entry = (second entry * third entry) / 2 would not hold. For example, with 16 we would get 16 * 18 / 2 = 144 which is not 108. Similarly, 18 * 18 / 2 = 162 and 20 * 18 / 2 = 180, both of which fail to match 108. Hence those options do not fit the discovered pattern.

Common Pitfalls:
Students sometimes look only at column wise relationships or simple differences and miss the more direct row wise pattern. Others might attempt addition based patterns instead of checking products. A systematic approach tries products, ratios, and sums row by row until a clear consistent rule emerges for all given rows.

Final Answer:
The number that should replace the question mark in the matrix is 12.