Alright, let's work through the problem step by step to find the missing value in the matrix.
We have the following 4x4 matrix:
[tex]\[ \begin{pmatrix}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3
\end{pmatrix} \][/tex]
Our goal is to find the missing value represented by ? in the fourth row.
### Step 1: Calculate the sums of the first three rows
First, let's compute the sum of each of the first three rows.
- Sum of the first row:
[tex]\[
6 + (-5) + (-6) + 5 = 6 - 5 - 6 + 5 = 0
\][/tex]
- Sum of the second row:
[tex]\[
-4 + 3 + 2 + (-6) = -4 + 3 + 2 - 6 = -5
\][/tex]
- Sum of the third row:
[tex]\[
6 + 6 + 9 + 4 = 6 + 6 + 9 + 4 = 25
\][/tex]
### Step 2: Determine the target sum for the fourth row
To estimate the missing value and understand the pattern, we assume the sum of the fourth row should match one of the row sums. Let's assume the sum of the fourth row should also be equal to the sum of the first row, which is 0. (This assumption may be arbitrary, based on the problem context or additional instructions not provided here.)
### Step 3: Set up the equation for the fourth row sum
The fourth row is: -9, ?, 6, 3.
Let's find the sum of the known elements in the fourth row:
[tex]\[
-9 + 6 + 3 = 0
\][/tex]
So, let's denote the missing value as [tex]\( x \)[/tex]. We form the equation for the sum of the fourth row:
[tex]\[
-9 + x + 6 + 3 = 0
\][/tex]
Simplifying:
[tex]\[
0 + x = 0
\][/tex]
### Step 4: Solve for the missing value
From the equation above:
[tex]\[
x = 0
\][/tex]
Therefore, the missing value [tex]\( ? \)[/tex] in the matrix is [tex]\( 0 \)[/tex]. Thus, the completed fourth row of the matrix is:
[tex]\[ \begin{pmatrix}
-9 & 0 & 6 & 3
\end{pmatrix} \][/tex]
So, the missing value is [tex]\( 0 \)[/tex].
