To find the number that completes the given pattern in the matrix, let's analyze the relationship between the numbers in each row. Each row appears to follow a specific algebraic relationship.
We'll look at the first three rows and identify a consistent pattern.
```
Row 1: 6, -5, -6, 5
Row 2: -4, 3, 2, -6
Row 3: 6, 6, 9, 4
```
First, consider the products of pairs of numbers in each row:
### Row 1:
- First two numbers: [tex]\( 6 \times (-5) = -30 \)[/tex]
- Last two numbers: [tex]\( -6 \times 5 = -30 \)[/tex]
The product of the first two numbers is equal to the product of the last two numbers.
### Row 2:
- First two numbers: [tex]\( -4 \times 3 = -12 \)[/tex]
- Last two numbers: [tex]\( 2 \times (-6) = -12 \)[/tex]
Again, the product of the first two numbers is equal to the product of the last two numbers.
### Row 3:
- First two numbers: [tex]\( 6 \times 6 = 36 \)[/tex]
- Last two numbers: [tex]\( 9 \times 4 = 36 \)[/tex]
Similarly, the product of the first two numbers is equal to the product of the last two numbers.
From these observations, we can deduce a pattern:
- For each row, the product of the first two numbers equals the product of the last two numbers.
### Row 4:
Given: [tex]\(-9, ? , 6, 3 \)[/tex]
We need to determine the value of [tex]\( ? \)[/tex] that satisfies the pattern. Let's denote the missing number as [tex]\( x \)[/tex].
Using the identified pattern:
- The product of the first two numbers should equal the product of the last two numbers.
Thus, the equation for the fourth row should be:
[tex]\[ -9 \times x = 6 \times 3 \][/tex]
Solving this equation:
[tex]\[ -9x = 18 \][/tex]
To isolate [tex]\( x \)[/tex], divide both sides of the equation by [tex]\(-9\)[/tex]:
[tex]\[ x = \frac{18}{-9} \][/tex]
[tex]\[ x = -2 \][/tex]
Therefore, the missing number that completes the pattern in the fourth row is [tex]\( \boxed{-2} \)[/tex].
