To solve for the value of [tex]\( d_{21} + d_{22} + d_{23} \)[/tex] given the matrix equation:
[tex]\[ D + \begin{bmatrix}
1 & 0 & 1 \\
-3 & 9 & -7 \\
-20 & 2 & 4
\end{bmatrix} = \begin{bmatrix}
9 & -3 & 2 \\
8 & -1 & 0 \\
11 & 15 & 3
\end{bmatrix} \][/tex]
we will follow several steps to isolate and calculate the elements of the matrix [tex]\( D \)[/tex].
### Step 1: Set up the equation for [tex]\( D \)[/tex]
The given matrix equation is:
[tex]\[ D + \begin{bmatrix}
1 & 0 & 1 \\
-3 & 9 & -7 \\
-20 & 2 & 4
\end{bmatrix} = \begin{bmatrix}
9 & -3 & 2 \\
8 & -1 & 0 \\
11 & 15 & 3
\end{bmatrix} \][/tex]
To isolate [tex]\( D \)[/tex], we subtract the matrix [tex]\(\begin{bmatrix}
1 & 0 & 1 \\
-3 & 9 & -7 \\
-20 & 2 & 4
\end{bmatrix}\)[/tex] from both sides of the equation:
[tex]\[ D = \begin{bmatrix}
9 & -3 & 2 \\
8 & -1 & 0 \\
11 & 15 & 3
\end{bmatrix} - \begin{bmatrix}
1 & 0 & 1 \\
-3 & 9 & -7 \\
-20 & 2 & 4
\end{bmatrix} \][/tex]
### Step 2: Perform the matrix subtraction
Subtract the corresponding elements of the matrices:
[tex]\[ D = \begin{bmatrix}
9 - 1 & -3 - 0 & 2 - 1 \\
8 - (-3) & -1 - 9 & 0 - (-7) \\
11 - (-20) & 15 - 2 & 3 - 4
\end{bmatrix} \][/tex]
### Step 3: Calculate the elements of [tex]\( D \)[/tex]
[tex]\[ D = \begin{bmatrix}
8 & -3 & 1 \\
11 & -10 & 7 \\
31 & 13 & -1
\end{bmatrix} \][/tex]
### Step 4: Find [tex]\( d_{21} \)[/tex], [tex]\( d_{22} \)[/tex], and [tex]\( d_{23} \)[/tex]
From the matrix [tex]\( D \)[/tex]:
[tex]\[ d_{21} = 11, \][/tex]
[tex]\[ d_{22} = -10, \][/tex]
[tex]\[ d_{23} = 7 \][/tex]
### Step 5: Calculate [tex]\( d_{21} + d_{22} + d_{23} \)[/tex]
Add the values of [tex]\( d_{21} \)[/tex], [tex]\( d_{22} \)[/tex], and [tex]\( d_{23} \)[/tex]:
[tex]\[ d_{21} + d_{22} + d_{23} = 11 + (-10) + 7 = 8 \][/tex]
### Answer
The value of [tex]\( d_{21} + d_{22} + d_{23} \)[/tex] is [tex]\( 8 \)[/tex].
