In the following question, a matrix of numbers is given. These numbers follow a particular numerical trend row wise or column wise. Study this trend carefully and choose the missing number accordingly. The matrix is: 1 7 9; 2 14 ?; 3 105 117. Which number should replace the question mark ?

Introduction / Context:
This is a three row numerical matrix puzzle with one missing value in the second row. Each column seems to connect its three elements through a multiplicative relationship. Our goal is to identify a rule that holds for all three columns, and then use that rule to compute the missing entry in the central position of the matrix.

Given Data / Assumptions:

• Row 1: 1, 7, 9.
• Row 2: 2, 14, ?.
• Row 3: 3, 105, 117.
• The missing number is the third element of the second row.
• The pattern is likely consistent down each column.
• We can use basic multiplication and addition to search for the rule.

Concept / Approach:
Rather than only looking row wise, we can treat each column as a mini sequence with three values. The top two values may be combined through multiplication and addition to generate the third value in the same column. By examining columns where all three entries are known, we can infer the relationship and apply it to the column that contains the missing value.

Step-by-Step Solution:
Step 1: Consider the first column: 1, 2, 3. The numbers 1 and 2 combine to form 3 by the equation 1 * 2 + 1 = 3. That is, third element = first element * second element + first element. Step 2: Now consider the second column: 7, 14, 105. Test the same pattern: 7 * 14 + 7 = 98 + 7 = 105. This matches the actual third element in the second column. Step 3: Thus, the rule third element = first element * second element + first element works for column 1 and column 2. Step 4: Apply this rule to the third column, where the first and third elements are known: 9 is the first element of the first row, the missing number is in the second row, and 117 is the third element in the same column. Step 5: Let the missing number be x. According to the rule, 117 = 9 * x + 9. Step 6: Solve for x. First subtract 9 from both sides: 117 − 9 = 108, so 9 * x = 108. Step 7: Divide both sides by 9: x = 108 / 9 = 12. Step 8: Hence, the missing number in the second row, third column is 12.

Verification / Alternative check:
We can quickly check consistency. For column 3 with values 9, 12, 117, applying the rule 9 * 12 + 9 gives 108 + 9 = 117, which matches the third entry. Since the same rule holds for all three columns, it confirms that we have identified the correct pattern and that 12 is the only number that fits.

Why Other Options Are Wrong:
If we substitute 26, 20, or 16 for the missing value, the rule 9 * x + 9 would give 243, 189, or 153 respectively instead of 117. None of these match the actual third column entry. Therefore those options do not satisfy the pattern that automatically holds for the first two columns.

Common Pitfalls:
Many candidates first look for a simple row wise progression and may miss a more structured column wise relationship. Others may try only addition or only multiplication rather than a combined form like first * second + first. Practising conversion between compact formulas and observed numeric patterns helps build confidence for such questions.

Final Answer:
The missing number in the matrix is 12.