Let's analyze the given matrix and find the value represented by the question mark (?). Here's the matrix:
[tex]\[
\begin{array}{cccc}
-4 & -6 & -9 & -6 \\
? & 7 & 4 & -4 \\
9 & 3 & 2 & 6 \\
3 & -4 & 8 & -6
\end{array}
\][/tex]
First, we find the sum of each row that has no missing values.
### Step 1: Calculate the sum of each complete row
1. First row:
[tex]\[
-4 + (-6) + (-9) + (-6) = -25
\][/tex]
2. Third row:
[tex]\[
9 + 3 + 2 + 6 = 20
\][/tex]
3. Fourth row:
[tex]\[
3 + (-4) + 8 + (-6) = 3 - 4 + 8 - 6 = 1
\][/tex]
Therefore, the sums of the rows are:
[tex]\[
\text{Row Sums} = [-25, 20, 1]
\][/tex]
### Step 2: Calculate the average sum of the complete rows
To find an average sum for the given rows, we add the row sums and divide by the number of rows:
[tex]\[
\text{Average Sum} = \frac{-25 + 20 + 1}{3} = \frac{-4}{3} = -1.3333
\][/tex]
### Step 3: Calculate the sum of the incomplete row (without the missing value)
For the second row, we have:
[tex]\[
7 + 4 + (-4) = 7
\][/tex]
### Step 4: Determine the missing value
We need to find a value for [tex]\( ? \)[/tex] that, when added to 7, gives us a total that brings the row sum closer to the average sum of [tex]\(-1.3333\)[/tex]. Therefore:
[tex]\[
? + 7 = -1.3333
\][/tex]
Solving for [tex]\( ? \)[/tex]:
[tex]\[
? = -1.3333 - 7 = -8.3333
\][/tex]
Thus, the value replacing the question mark [tex]\( ? \)[/tex] is approximately:
[tex]\[
-8.3333
\][/tex]
