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Determine the missing value in the matrix.

[tex]\[ \begin{array}{rrrr}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3
\end{array} \][/tex]


Sagot :

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]