Let's break down the given problem step-by-step to find the missing value (?).
We are given a matrix:
[tex]\[
\begin{array}{rrrr}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3 \\
\end{array}
\][/tex]
Firstly, we need to calculate the sum for each known column:
1. Sum of the 1st column:
[tex]\[
6 + (-4) + 6 + (-9) = 6 - 4 + 6 - 9 = -1
\][/tex]
2. Sum of the 3rd column:
[tex]\[
-6 + 2 + 9 + 6 = -6 + 2 + 9 + 6 = 11
\][/tex]
3. Sum of the 4th column:
[tex]\[
5 + (-6) + 4 + 3 = 5 - 6 + 4 + 3 = 6
\][/tex]
Next, let's calculate the partial sum of the known values in the 2nd column without including the unknown value (?):
[tex]\[
-5 + 3 + 6 = -5 + 3 + 6 = 4
\][/tex]
We denote the total sum of the second column as [tex]\(S_2\)[/tex]. Our goal is to find the missing value (?), which we denote as [tex]\(x\)[/tex].
The sum of all elements considering the unknown value in the second column is:
Using the total sum of known partial sums of other columns:
[tex]\[
\text{Sum of 1st column} + \text{Sum of 3rd column} + \text{Sum of 4th column} = -1 + 11 + 6 = 16
\][/tex]
Since the sum of the 2nd column must balance the entire array. We are given a balance condition, which gives a final column sum to be 0 including the unknown. This would be written as:
[tex]\[
4 + x = 0
\][/tex]
So, we solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4
\][/tex]
So, the missing value in the matrix is [tex]\(-4\)[/tex]. Summarizing our values:
- Sum of the 1st column: [tex]\(-1\)[/tex]
- Sum of the 3rd column: [tex]\(11\)[/tex]
- Sum of the 4th column: [tex]\(6\)[/tex]
- Partial sum of the 2nd column excluding the unknown value: [tex]\(4\)[/tex]
- The missing value [tex]\( ? = -4 \)[/tex]
