To determine the missing value in the provided matrix, we'll analyze potential patterns in the differences between consecutive elements in columns. We will also consider the sum of elements in rows.
Given matrix:
[tex]\[ \begin{array}{cccc}
9 & 1 & 6 & 4 \\
4 & 5 & 7 & 2 \\
5 & 8 & 8 & 5 \\
1 & 3 & 5 & ? \\
\end{array} \][/tex]
### Step-by-Step Analysis:
1. Column-wise Differences:
Column 1 (First Column):
[tex]\[
\begin{align*}
\text{Difference between 4 and 9} &= 4 - 9 = -5 \\
\text{Difference between 5 and 4} &= 5 - 4 = 1 \\
\text{Difference between 1 and 5} &= 1 - 5 = -4 \\
\end{align*}
\][/tex]
The differences are: [tex]\(-5, 1, -4\)[/tex].
Column 2 (Second Column):
[tex]\[
\begin{align*}
\text{Difference between 5 and 1} &= 5 - 1 = 4 \\
\text{Difference between 8 and 5} &= 8 - 5 = 3 \\
\text{Difference between 3 and 8} &= 3 - 8 = -5 \\
\end{align*}
\][/tex]
The differences are: [tex]\(4, 3, -5\)[/tex].
Column 3 (Third Column):
[tex]\[
\begin{align*}
\text{Difference between 7 and 6} &= 7 - 6 = 1 \\
\text{Difference between 8 and 7} &= 8 - 7 = 1 \\
\text{Difference between 5 and 8} &= 5 - 8 = -3 \\
\end{align*}
\][/tex]
The differences are: [tex]\(1, 1, -3\)[/tex].
2. Row Sum Calculation:
Let's analyze the rows to find the sum. The sum of the known elements of the first three rows are:
[tex]\[
\begin{align*}
\text{Sum of Row 1} &= 9 + 1 + 6 + 4 = 20 \\
\text{Sum of Row 2} &= 4 + 5 + 7 + 2 = 18 \\
\text{Sum of Row 3} &= 5 + 8 + 8 + 5 = 26 \\
\end{align*}
\][/tex]
The total sum of the first three rows is:
[tex]\[
20 + 18 + 26 = 64
\][/tex]
Next, we sum the known elements of Row 4:
[tex]\[
1 + 3 + 5 = 9
\][/tex]
To balance the rows, we need the missing value such that the total sum of Row 4 equals the combined sum of the other rows. Here’s the missing value calculation:
[tex]\[
? = 64 - 9 = 55
\][/tex]
Therefore, the missing value [tex]\("?") is \(55\)[/tex].
