Short Answer 
The process involves analyzing patterns in matrix rows by calculating the products of paired numbers, confirming that the product of the first two equals the product of the last two for each row. By applying this rule to a fourth row with a missing number, the missing value is determined to be -2.
Step 1: Analyze the Row Patterns
Begin by examining the provided matrix rows to identify a consistent pattern. For each row, observe the relationship between the numbers and note how they interact through multiplication. Analyze the product of the numbers in pairs, focusing on how these products compare across each row.
- Row 1: 6, -5, -6, 5
- Row 2: -4, 3, 2, -6
- Row 3: 6, 6, 9, 4
Step 2: Identify the Multiplicative Consistency
For each row, calculate the products of the first two numbers and the last two numbers to establish a rule. Notice that each row follows the format where the product of the first two numbers equals the product of the last two numbers. This rule will be critical for finding the missing element in the fourth row.
- Row 1: 6 √ó (-5) = -30 and -6 √ó 5 = -30
- Row 2: -4 √ó 3 = -12 and 2 √ó (-6) = -12
- Row 3: 6 √ó 6 = 36 and 9 √ó 4 = 36
Step 3: Solve for the Missing Number
Using the identified pattern, set up an equation for the fourth row where one number is missing. Denote the missing number as x. Apply the established rule that the product of the first two numbers should equal the product of the last two numbers to solve for x.
- Equation: -9 √ó x = 6 √ó 3
- Simplifying gives: -9x = 18
- Solving for x gives: x = -2
Thus, the missing number that completes the pattern is -2.