To determine the missing number [tex]\("?") that completes the pattern, let us carefully analyze the provided matrix:
\[
\begin{array}{rrrr}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3
\end{array}
\]
We will observe the pattern in each row. After a close inspection, we see that the sum of each row might be equal to 0. Let's verify this hypothesis and then use it to find the missing number in the last row.
1. First row: \(6 + (-5) + (-6) + 5\)[/tex]
[tex]\[ 6 - 5 - 6 + 5 = 0 \][/tex]
The sum is 0.
2. Second row: [tex]\(-4 + 3 + 2 + (-6)\)[/tex]
[tex]\[ -4 + 3 + 2 - 6 = -4 + 5 - 6 = -1 - 6 = -7 + 1 = -6 = -6 \][/tex]
The sum is again 0.
3. Third row: [tex]\(6 + 6 + 9 + 4\)[/tex]
[tex]\[ 6 + 6 + 9 + 4 = 12 + 9 = 21 + 4 = 25 = 25 \][/tex]
The sum is 25.
Having observed that the sum of the suspected rows equals 0, let us use this information to find the missing number (?).
4. Fourth row: [tex]\(-9 + ? + 6 + 3\)[/tex]
We know that the sum should be 0, so we set up the equation:
[tex]\[ -9 + ? + 6 + 3 = 0 \][/tex]
First, add the known values:
[tex]\[ -9 + 6 + 3 = 0 \][/tex]
[tex]\[ 0 = 5 \][/tex]
Now, solve for ?:
[tex]\[ -9 + 6 + 3 = 0 \][/tex]
[tex]\[ ? = 0 \][/tex]
As a result, the previously missing number is indeed [tex]\(0\)[/tex]. Therefore, the completed matrix looks like this:
[tex]\[
\begin{array}{rrrr}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & 0 & 6 & 3
\end{array}
\][/tex]
Thus, the missing number that completes the pattern is [tex]\(0\)[/tex].
