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We have the following matrices:

\[
A = \begin{bmatrix} 1 & -d \\ b & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -5 \\ -3 & b \end{bmatrix}, \quad C = \begin{bmatrix} 3c & -5c \\ 1-a & 3a-1 \end{bmatrix}
\]

The problem asks to find the value of \( a + b + c + d \) given that:

\[
A + 2B^T = C
\]

First, let's compute the transpose of matrix \( B \) (denoted as \( B^T \)):

\[
B^T = \begin{bmatrix} 4 & -3 \\ -5 & b \end{bmatrix}
\]

Next, we compute \( 2B^T \):

\[
2B^T = 2 \times \begin{bmatrix} 4 & -3 \\ -5 & b \end{bmatrix} = \begin{bmatrix} 8 & -6 \\ -10 & 2b \end{bmatrix}
\]

Now, let's find \( A + 2B^T \):

\[
A + 2B^T = \begin{bmatrix} 1 & -d \\ b & 3 \end{bmatrix} + \begin{bmatrix} 8 & -6 \\ -10 & 2b \end{bmatrix} = \begin{bmatrix} 9 & -d-6 \\ b-10 & 3+2b \end{bmatrix}
\]

Given that \( A + 2B^T = C \), we equate this result with matrix \( C \):

\[
\begin{bmatrix} 9 & -d-6 \\ b-10 & 3+2b \end{bmatrix} = \begin{bmatrix} 3c & -5c \\ 1-a & 3a-1 \end{bmatrix}
\]

This gives us the following system of equations:

1. \( 9 = 3c \) 
2. \( -d - 6 = -5c \)
3. \( b - 10 = 1 - a \)
4. \( 3 + 2b = 3a - 1 \)

Let's solve these step-by-step:

1. From the first equation \( 9 = 3c \):
   \[
   c = 3
   \]

2. From the second equation \( -d - 6 = -5c \):
   \[
   -d - 6 = -5 \times 3 \\
   -d - 6 = -15 \\
   -d = -15 + 6 \\
   d = 9
   \]

3. From the third equation \( b - 10 = 1 - a \):
   \[
   b = 1 - a + 10 \\
   b = 11 - a
   \]

4. Substitute \( b = 11 - a \) into the fourth equation \( 3 + 2b = 3a - 1 \):
   \[
   3 + 2(11 - a) = 3a - 1 \\
   3 + 22 - 2a = 3a - 1 \\
   25 - 2a = 3a - 1 \\
   25 + 1 = 3a + 2a \\
   26 = 5a \\
   a = \frac{26}{5}
   \]

5. Using \( a = \frac{26}{5} \), we find \( b \):
   \[
   b = 11 - a = 11 - \frac{26}{5} = \frac{55}{5} - \frac{26}{5} = \frac{29}{5}
   \]

Finally, we compute \( a + b + c + d \):

\[
a + b + c + d = \frac{26}{5} + \frac{29}{5} + 3 + 9 = \frac{26 + 29}{5} + 12 = \frac{55}{5} + 12 = 11 + 12 = 23
\]

So, the value of \( a + b + c + d \) is \( \boxed{23} \).

Do you want details or have any questions? Here are five related questions:

1. How do you find the transpose of a matrix?
2. What is the process for adding two matrices?
3. How do you multiply a matrix by a scalar?
4. Can you explain the general method to equate two matrices?
5. What are some applications of matrix operations in real life?

**Tip:** Always double-check each step in a matrix operation, especially sign changes and arithmetic.

Diketahui matriks A = [1 -d; b 3], B = [4 -5; -3 b], dan C = [3c -5c; 1 - a 3a - 1]. Jika A + 2B^T = C, tentukan nilai a + b + c + d!

This problem involves matrix operations, including transpose, scalar multiplication, and solving systems of equations. Given matrices A, B, and C, the task is to find a + b + c + d by equating A + 2B^T to matrix C.

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