To find the missing number represented by the symbol "[tex]$?$[/tex]" in the given matrix, we can follow these steps:
1. Identify the matrix:
[tex]\[
\begin{array}{cccc}
9 & 1 & 6 & 4 \\
4 & 5 & 7 & 2 \\
5 & 8 & 8 & 5 \\
1 & 3 & 5 & ? \\
\end{array}
\][/tex]
2. Analyze the sums of each row:
- The sum of the first row: [tex]\(9 + 1 + 6 + 4 = 20\)[/tex]
- The sum of the second row: [tex]\(4 + 5 + 7 + 2 = 18\)[/tex]
- The sum of the third row: [tex]\(5 + 8 + 8 + 5 = 26\)[/tex]
- The sum of the fourth row (excluding the missing number): [tex]\(1 + 3 + 5 = 9\)[/tex]
3. Determine the pattern:
- By inspecting the sums of the rows, it appears that there isn’t an immediately obvious consistent pattern for all sums of the rows. For a more consistent approach, one method is to use the sum of a known complete row as a reference.
4. Select a target row sum:
- Considering the third row’s sum is [tex]\(26\)[/tex]:
[tex]\[
5 + 8 + 8 + 5 = 26
\][/tex]
5. Calculate the missing number:
- Using the target sum (26) from the third row, we aim for the fourth row's sum to also be [tex]\(26\)[/tex]:
[tex]\[
1 + 3 + 5 + ? = 26
\][/tex]
- Let [tex]\( ? \)[/tex] be the missing number.
[tex]\[
1 + 3 + 5 + ? = 26
\][/tex]
- Sum of the known numbers in the fourth row:
[tex]\[
1 + 3 + 5 = 9
\][/tex]
- Solve for the missing number:
[tex]\[
9 + ? = 26
\][/tex]
[tex]\[
? = 26 - 9
\][/tex]
[tex]\[
? = 17
\][/tex]
6. Conclusion:
- Therefore, the missing number in the matrix is [tex]\( \boxed{17} \)[/tex].
The filled matrix looks as follows:
[tex]\[
\begin{array}{cccc}
9 & 1 & 6 & 4 \\
4 & 5 & 7 & 2 \\
5 & 8 & 8 & 5 \\
1 & 3 & 5 & 17 \\
\end{array}
\][/tex]
