Let's find the matrix [tex]\( X \)[/tex] that verifies the equation
[tex]\[ 2X + 3(A + B) = CD \][/tex]
where
[tex]\[ A = \begin{pmatrix} -1 & 3 \\ 6 & 8 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 3 \\ -1 & 5 \end{pmatrix}, \quad C = \begin{pmatrix} 4 & 2 & 1 \\ 3 & -2 & 3 \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 2 \\ -2 & 4 \\ -1 & 6 \end{array} \][/tex]
### Step 1: Calculate [tex]\(A + B\)[/tex]
First, we add matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{pmatrix} -1 & 3 \\ 6 & 8 \end{pmatrix} + \begin{pmatrix} 2 & 3 \\ -1 & 5 \end{pmatrix} = \begin{pmatrix} -1+2 & 3+3 \\ 6-1 & 8+5 \end{pmatrix} = \begin{pmatrix} 1 & 6 \\ 5 & 13 \end{pmatrix} \][/tex]
### Step 2: Calculate [tex]\(3(A + B)\)[/tex]
Next, we multiply [tex]\( A + B \)[/tex] by 3:
[tex]\[ 3(A + B) = 3 \cdot \begin{pmatrix} 1 & 6 \\ 5 & 13 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 & 3 \cdot 6 \\ 3 \cdot 5 & 3 \cdot 13 \end{pmatrix} = \begin{pmatrix} 3 & 18 \\ 15 & 39 \end{pmatrix} \][/tex]
### Step 3: Calculate [tex]\(CD\)[/tex]
Now, we compute the matrix product [tex]\( CD \)[/tex]:
[tex]\[ C = \begin{pmatrix} 4 & 2 & 1 \\ 3 & -2 & 3 \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 2 \\ -2 & 4 \\ -1 & 6 \end{pmatrix} \][/tex]
[tex]\[ CD = \begin{pmatrix}
(4 \cdot 1 + 2 \cdot (-2) + 1 \cdot (-1)) & (4 \cdot 2 + 2 \cdot 4 + 1 \cdot 6) \\
(3 \cdot 1 + (-2) \cdot (-2) + 3 \cdot (-1)) & (3 \cdot 2 + (-2) \cdot 4 + 3 \cdot 6)
\end{pmatrix} = \begin{pmatrix}
-1 & 22 \\
4 & 16
\end{pmatrix} \][/tex]
### Step 4: Substitute into the Equation
The equation to solve for [tex]\( X \)[/tex] is:
[tex]\[ 2X + 3(A + B) = CD \][/tex]
From steps 2 and 3, we have:
[tex]\[ 2X + \begin{pmatrix} 3 & 18 \\ 15 & 39 \end{pmatrix} = \begin{pmatrix} -1 & 22 \\ 4 & 16 \end{pmatrix} \][/tex]
### Step 5: Rearrange to Solve for [tex]\( 2X \)[/tex]
Subtract [tex]\( 3(A + B) \)[/tex] from both sides:
[tex]\[ 2X = CD - 3(A + B) \][/tex]
[tex]\[ 2X = \begin{pmatrix} -1 & 22 \\ 4 & 16 \end{pmatrix} - \begin{pmatrix} 3 & 18 \\ 15 & 39 \end{pmatrix} = \begin{pmatrix} -1-3 & 22-18 \\ 4-15 & 16-39 \end{pmatrix} = \begin{pmatrix} -4 & 4 \\ -11 & -23 \end{pmatrix} \][/tex]
### Step 6: Solve for [tex]\( X \)[/tex]
Divide both sides of the equation by 2:
[tex]\[ X = \frac{1}{2} \begin{pmatrix} -4 & 4 \\ -11 & -23 \end{pmatrix} \][/tex]
[tex]\[ X = \begin{pmatrix} -2 & 2 \\ -5.5 & -11.5 \end{pmatrix} \][/tex]
Thus, the matrix [tex]\( X \)[/tex] that satisfies the equation is:
[tex]\[ X = \begin{pmatrix} -2 & 2 \\ -5.5 & -11.5 \end{pmatrix} \][/tex]
