Certainly! Let's solve the given matrix problem step-by-step.
We are given a matrix:
[tex]\[
\begin{array}{rrrr}
2 & -3 & 9 & 5 \\
8 & ? & 3 & 7 \\
-4 & 2 & 4 & -6 \\
5 & 2 & 8 & 1
\end{array}
\][/tex]
where one value (denoted as ?) is missing in the second column. Our goal is to find this missing value by following these steps:
### Step 1: Identify the Missing Value Position
The missing value is located at the position of the second row and second column.
### Step 2: Calculate the Mean of the Second Column
To fill the missing value, we will calculate the mean of the existing values in the second column, ignoring the missing value. The second column contains:
[tex]\[
\begin{array}{c}
-3 \\
? \\
2 \\
2
\end{array}
\][/tex]
We'll calculate the mean of the existing values: [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(2\)[/tex].
The formula for the arithmetic mean is:
[tex]\[
\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}
\][/tex]
[tex]\[
\text{Mean} = \frac{-3 + 2 + 2}{3} = \frac{1}{3} \approx 0.3333
\][/tex]
So, the mean of the second column is approximately [tex]\(0.3333\)[/tex].
### Step 3: Fill the Missing Value with the Calculated Mean
Now we will replace the missing value with the calculated mean:
[tex]\[
\text{Missing value} = 0.3333
\][/tex]
### Step 4: Update the Matrix
We can now update the matrix with the calculated mean to get:
[tex]\[
\begin{array}{rrrr}
2 & -3 & 9 & 5 \\
8 & 0.3333 & 3 & 7 \\
-4 & 2 & 4 & -6 \\
5 & 2 & 8 & 1
\end{array}
\][/tex]
Therefore, the missing value in the second column has been filled with approximately [tex]\(0.3333\)[/tex], and the updated matrix is:
[tex]\[
\begin{array}{rrrr}
2 & -3 & 9 & 5 \\
8 & 0.3333 & 3 & 7 \\
-4 & 2 & 4 & -6 \\
5 & 2 & 8 & 1
\end{array}
\][/tex]
