To determine the missing value in the matrix, follow the steps below:
We are given a 4x4 matrix with one element missing. The last element in the matrix needs to be found:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
7 & 2 & 1 & 9 \\
\hline
8 & 5 & 0 & 3 \\
\hline
4 & 3 & 1 & 7 \\
\hline
5 & 2 & 0 & ? \\
\hline
\end{array}
\][/tex]
First, observe the matrix pattern to infer the missing value. Consider the sum of all elements in each row.
Calculate the sum of elements in the rows that already have all elements:
1. First row: [tex]\(7 + 2 + 1 + 9 = 19\)[/tex].
2. Second row: [tex]\(8 + 5 + 0 + 3 = 16\)[/tex].
3. Third row: [tex]\(4 + 3 + 1 + 7 = 15\)[/tex].
Now consider the partially completed fourth row. Summing the known elements in this row:
[tex]\[
5 + 2 + 0 = 7
\][/tex]
To maintain a consistent pattern in the sums of the rows, suppose that each row's sum should ideally be equal or follow a progression. The sums of the first, second, and third rows are 19, 16, and 15, respectively. This sequence does not show an increasing or consistent pattern; therefore, the explanation may not rely purely on sums.
Another plausible method is to consider whether the missing value balances out with the column sums or matches other elements criteria. However, without an apparent pattern or additional rule, a likely acceptable reasoning strategy is via extending typical properties.
For a simple approach based on sum patterns, calculate the missing element such that:
Given sum parts (19, 16, 15), the simplest row sum assumption can be deduced \(x=8 \implies 7 + x = the balancing \text{ of consistency or higher likely fairness}).
So the missing cell value:
[tex]\[ 5 + 2 + 0 + x = 15 \][/tex]
[tex]\[ 15 - 7 \][/tex]
[tex]\[ 15 - 7 = 8 \][/tex]
Therefore, the missing number is 8.
So, the complete matrix would be:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
7 & 2 & 1 & 9 \\
\hline
8 & 5 & 0 & 3 \\
\hline
4 & 3 & 1 & 7 \\
\hline
5 & 2 & 0 & 8 \\
\hline
\end{array}
\][/tex]
