To determine the missing number in the given matrix where an arithmetic pattern exists, we'll analyze existing patterns within each column and apply them to determine the missing number.
First, let’s observe and analyze each column:
Column 1:
[tex]\[ \begin{array}{r}
6 \\
-4 \\
6 \\
-9
\end{array} \][/tex]
1. From 6 to -4, the change is [tex]\( -4 - 6 = -10 \)[/tex].
2. From -4 to 6, the change is [tex]\( 6 - (-4) = 10 \)[/tex].
3. From 6 to -9, the change is [tex]\( -9 - 6 = -15 \)[/tex].
The changes do not form a clear arithmetic sequence, so we check other columns.
Column 2:
[tex]\[ \begin{array}{r}
-5 \\
3 \\
6 \\
?
\end{array} \][/tex]
1. From -5 to 3, the change is [tex]\( 3 - (-5) = 8 \)[/tex].
2. From 3 to 6, the change is [tex]\( 6 - 3 = 3 \)[/tex].
Based on these changes:
- The change from the first element to the second is [tex]\( +8 \)[/tex].
- The change from the second element to the third is [tex]\( +3 \)[/tex].
We will test consistency in Column 3.
Column 3:
[tex]\[ \begin{array}{r}
-6 \\
2 \\
9 \\
6
\end{array} \][/tex]
1. From -6 to 2, the change is [tex]\( 2 - (-6) = 8 \)[/tex].
2. From 2 to 9, the change is [tex]\( 9 - 2 = 7 \)[/tex].
3. From 9 to 6, the change is [tex]\( 6 - 9 = -3 \)[/tex].
In Column 3, the changes [tex]\( (8, 7, -3) \)[/tex] do not form a clear arithmetic sequence either.
Finally, we check Column 4's consistency.
Column 4:
[tex]\[ \begin{array}{r}
5 \\
-6 \\
4 \\
3
\end{array} \][/tex]
1. From 5 to -6, the change is [tex]\( -6 - 5 = -11 \)[/tex].
2. From -6 to 4, the change is [tex]\( 4 - (-6) = 10 \)[/tex].
3. From 4 to 3, the change is [tex]\( 3 - 4 = -1 \)[/tex].
We see varying changes in columns 1, 3, and 4.
Observing Column 2 Page comprehensively suggests the progression:
- (-5 to 3 is +8)
- (3 to 6 is +3)
If such a pattern (+8, +3) is applied, we hypothesize a potential:
- Last part of column 2: [tex]\( 6 + 3 = 9 \)[/tex].
Thus, the missing number [tex]\( ? = 9 \)[/tex].
So, the answer is:
[tex]\( 9 \)[/tex]
