To solve the problem of finding the missing element in the matrix, we proceed as follows:
Given matrix:
[tex]\[
\begin{array}{cccc}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3
\end{array}
\][/tex]
### Step 1: Calculate the sum of each column excluding the missing element.
First, we calculate the sum for each column, ignoring the missing value:
1. For the first column:
[tex]\[
6 + (-4) + 6 + (-9) = 6 - 4 + 6 - 9 = -1
\][/tex]
2. For the second column (ignoring the missing value `?`):
[tex]\[
-5 + 3 + 6 = 4
\][/tex]
3. For the third column:
[tex]\[
-6 + 2 + 9 + 6 = -6 + 2 + 9 + 6 = 11
\][/tex]
4. For the fourth column:
[tex]\[
5 - 6 + 4 + 3 = 5 - 6 + 4 + 3 = 6
\][/tex]
So, the sums of the columns (excluding the missing element) are:
[tex]\[
[-1, 4, 11, 6]
\][/tex]
### Step 2: Identify the consistent column sum.
The sum of the first column is -1, and we will use this as the reference (expected) sum for the other columns.
### Step 3: Calculate the sum of elements in the column with the missing value.
The sum of the second column (excluding the missing value) is:
[tex]\[
4
\][/tex]
### Step 4: Determine the missing value to balance the column sum.
We need the sum of the second column to be -1 (same as the reference sum from the first column). Therefore, we need to find the value of `?` that will balance the column sum to -1.
From the sum we have:
[tex]\[
4 + ? = -1
\][/tex]
Solving for `?`:
[tex]\[
? = -1 - 4 = -5
\][/tex]
Thus, the missing value in the matrix is [tex]\(-5\)[/tex].
### Conclusion
The missing value is [tex]\(-5\)[/tex]. Therefore, the populated matrix is:
[tex]\[
\begin{array}{cccc}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & -5 & 6 & 3
\end{array}
\][/tex]
