Let's solve the problem step by step by first understanding the given matrix and what information we need to find. We have:
[tex]\[
\begin{array}{|c|c|c|}
\hline
16 & 28 & 29 \\
\hline
13 & 12 & 16 \\
\hline
14 & 10 & 15 \\
\hline
15 & 30 & ? \\
\hline
\end{array}
\][/tex]
Our objective is to find the missing number in the last entry of the third column, denoted as [tex]\( ? \)[/tex].
First, let's calculate the sum of each column:
[tex]\[
\begin{aligned}
&\text{Sum of Column 1: } 16 + 13 + 14 + 15, \\
&\text{Sum of Column 2: } 28 + 12 + 10 + 30,\\
&\text{Sum of Column 3: } 29 + 16 + 15 + ?.
\end{aligned}
\][/tex]
Let's compute these sums step-by-step.
1. Sum of Column 1:
[tex]\[
16 + 13 + 14 + 15 = 58.
\][/tex]
2. Sum of Column 2:
[tex]\[
28 + 12 + 10 + 30 = 80.
\][/tex]
3. Sum of Column 3 without the missing value:
[tex]\[
29 + 16 + 15 = 60.
\][/tex]
We assume that the sum of each column should be approximately equal (within a reasonable range since no other pattern or information is given). To estimate the missing number, we can find the average of the sums of the two known columns.
[tex]\[
\text{Average sum of the known columns} = \frac{(58 + 80)}{2} = \frac{138}{2} = 69.
\][/tex]
We expect the sum of the third column to be 69. Therefore, we can find the missing number ([tex]\( ? \)[/tex]) by subtracting the sum of the known entries in the third column from this expected sum:
[tex]\[
? = 69 - 60 = 9.
\][/tex]
Thus, the missing number in the third column is [tex]\( \boxed{9} \)[/tex].
