1. The problem is to find the missing number represented by ? in the matrix: $$\begin{matrix}7 & ? & 108 \\ 8 & 12 & 96 \\ 9 & 132 & 84\end{matrix}$$ 2. Let's analyze the pattern in each row. - Row 2: 8, 12, 96 Check if the product of the first two numbers relates to the third: $$8 \times 12 = 96$$, matching the third number. - Row 3: 9, 132, 84 Check product of first two numbers: $$9 \times 132 = 1188$$, which does not equal 84. Let's check division: $$132 \div 9 = 14.67\ldots$$ $$84 \div 132 = 0.636\ldots$$ No straightforward pattern there. 3. Let's check the products of the second and third numbers in the rows: - Row 1: ? and 108 - Row 2: 12 and 96 - Row 3: 132 and 84 Investigate relationship between first and product of last two numbers: - Row 2: $$12 \times 96 = 1152$$ $$1152 \div 8 = 144$$ - Row 3: $$132 \times 84 = 11088$$ $$11088 \div 9 = 1232$$ - Row 1: Let ? = x $$x \times 108 = y$$ To find if y divided by 7 fits a pattern. Looking at these, maybe no clear pattern here, let's try sums. 4. Check sums before and after: - Row 2 sum: $$8 + 12 = 20$$ The third number is 96, no relation with 20. - Row 3 sum: $$9 + 132 = 141$$ Third number is 84. No clear product or sum pattern. 5. Check columns for patterns: - Column 1: 7, 8, 9 - Column 2: ?, 12, 132 - Column 3: 108, 96, 84 Notice that column 3 numbers decrease by 12 each row: $$108 - 12 = 96$$ $$96 - 12 = 84$$ Maybe column 2 is increasing similar way: From 12 to 132 increased by 120. If ? follows a pattern starting with 12 and jumps 120, maybe ? is -108 (12 - 120), but this is negative and unlikely. 6. Check ratio in column 3: $$108 \div 7 = 15.43$$ $$96 \div 8 = 12$$ $$84 \div 9 = 9.33$$ No clear sequence. 7. Let's check if multiplication of column 1 and column 2 equals column 3: - Row 2: $$8 \times 12 = 96$$ correct. - Row 3: $$9 \times 132 = 1188$$ which is not 84. So this doesn't hold. 8. Let's try multiplying column 1 by column 3 and dividing by column 2: - Row 2: $$8 \times 96 \div 12 = 64$$ - Row 3: $$9 \times 84 \div 132 = 5.73$$ No clear pattern. 9. Alternatively, observe column 2 entries: 12 and 132, possibly related by factor 11 (12 x 11 = 132). Try applying factor 11 again to find ?: If 7 x 11 = 77, so possibly ? = 77. Test row 1 with ?=77: Check if 7 x 77 = 108, no (7 x 77 = 539). But row 2 8 x 12 = 96 holds. No conclusive pattern. 10. Considering the consistency in Row 2, where 8 x 12 = 96. Assuming similar for Row 1, 7 x ? = 108. Solve for ?: $$? = \frac{108}{7} = 15.43$$ Since ? is likely an integer, check for nearest integer: Check 15 x 7 = 105 (close to 108). Check 16 x 7 = 112 (also close). Given lack of other patterns, the most reasonable answer is: $$\boxed{15.43}$$ or approximately 15. Hence the missing number ? is approximately 15.43.