Sure! Let's solve this step-by-step.
We are given a 4x4 matrix with a missing value denoted by "?". To find the missing value, we can use the information present in the matrix. Here is the matrix again:
[tex]\[
\begin{array}{cccc}
9 & 1 & 6 & 4 \\
4 & 5 & 7 & 2 \\
5 & 8 & 8 & 5 \\
1 & 3 & 5 & ? \\
\end{array}
\][/tex]
First, let's calculate the sum of the values in each column.
### Column Sums:
- Column 1: [tex]\(9 + 4 + 5 + 1 = 19\)[/tex]
- Column 2: [tex]\(1 + 5 + 8 + 3 = 17\)[/tex]
- Column 3: [tex]\(6 + 7 + 8 + 5 = 26\)[/tex]
- Column 4: [tex]\(4 + 2 + 5 + ? = 11 + ?\)[/tex]
We have the partial sums of the first three columns as follows:
- Sum of Column 1: [tex]\(19\)[/tex]
- Sum of Column 2: [tex]\(17\)[/tex]
- Sum of Column 3: [tex]\(26\)[/tex]
To find the sum of the elements in the fourth column, we need to assume that the total desired sum for each column is consistent. Since there are different sums, for simplicity, we can assume a target sum to find the missing value. Let’s assume each column should sum up to 34 because the sums of the other columns roughly lead us into believing the target should be around that number when adding the remaining values.
Sum up the provided column totals:
[tex]\[ 19 + 17 + 26 = 62 \][/tex]
From the assumption, if each column should sum to 34:
[tex]\[ 4 \times 34 = 136 \][/tex]
Then the sum of the fourth column can be obtained by subtracting the sum of the first three columns from our total column target sum of [tex]\(136\)[/tex]:
[tex]\[ 136 - 62 = 74 \][/tex]
Hence, the sum of the fourth column should be:
[tex]\[ 11 + ? = 34 \][/tex]
[tex]\[ ? = 34 - 11 \][/tex]
[tex]\[ ? = 23 \][/tex]
So the missing value is 23.
### Matrix with the missing value filled:
[tex]\[
\begin{array}{cccc}
9 & 1 & 6 & 4 \\
4 & 5 & 7 & 2 \\
5 & 8 & 8 & 5 \\
1 & 3 & 5 & 23 \\
\end{array}
\][/tex]
Thus, the missing value is [tex]\(23\)[/tex].
