https://file.aiquickdraw.com/mathbot/file/1727225173929nh8engg1.jpg

The image contains a matrix equation that reads:

\[
2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 1 & 3 \end{pmatrix}
\]

We are tasked with finding the values of \( a + b + c + d \).

### Step 1: Simplifying the Left Side
First, expand the matrices on the left-hand side.

1. Multiply the matrix \( \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} \) by 2:
   \[
   2 \begin{pmatrix} a & 2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix}
   \]

2. Add the matrix \( \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} \) to the result:
   \[
   \begin{pmatrix} 2a & 4 \\ -6 & 2 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 2a + 4 & 4 - 1 \\ -6 + 0 & 2 + b \end{pmatrix} = \begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix}
   \]

### Step 2: Simplifying the Right Side
Now, perform matrix multiplication on the right-hand side:
\[
\begin{pmatrix} 3 & 2 \\ c & 4 \end{pmatrix} \begin{pmatrix} 2 & d \\ 1 & 3 \end{pmatrix}
\]
Multiply the matrices:

\[
\begin{pmatrix} (3 \times 2 + 2 \times 1) & (3 \times d + 2 \times 3) \\ (c \times 2 + 4 \times 1) & (c \times d + 4 \times 3) \end{pmatrix} = \begin{pmatrix} 6 + 2 & 3d + 6 \\ 2c + 4 & cd + 12 \end{pmatrix} = \begin{pmatrix} 8 & 3d + 6 \\ 2c + 4 & cd + 12 \end{pmatrix}
\]

### Step 3: Equating Both Sides
We now have the following equation:

\[
\begin{pmatrix} 2a + 4 & 3 \\ -6 & 2 + b \end{pmatrix} = \begin{pmatrix} 8 & 3d + 6 \\ 2c + 4 & cd + 12 \end{pmatrix}
\]

By comparing the elements of both matrices:

1. \( 2a + 4 = 8 \) → \( 2a = 4 \) → \( a = 2 \)
2. \( 3 = 3d + 6 \) → \( 3d = -3 \) → \( d = -1 \)
3. \( -6 = 2c + 4 \) → \( 2c = -10 \) → \( c = -5 \)
4. \( 2 + b = cd + 12 = (-5)(-1) + 12 = 5 + 12 = 17 \) → \( b = 15 \)

### Step 4: Calculating \( a + b + c + d \)
Now, we calculate:
\[
a + b + c + d = 2 + 15 + (-5) + (-1) = 2 + 15 - 5 - 1 = 11
\]

Thus, the value of \( a + b + c + d \) is **11**.

---

Would you like further explanation on any steps? Here are 5 additional questions related to this:

1. How do you multiply two matrices step by step?
2. Why does matrix addition require the same dimensions for both matrices?
3. What is the determinant of a 2x2 matrix?
4. How would this problem change if the matrices were 3x3?
5. Can matrix multiplication be commutative, and if not, why?

**Tip:** Always double-check matrix dimensions before attempting operations to avoid errors.

Diketahui persamaan matriks: 2(a 2; -3 1) + (4 -1; 0 b) = (3 2; c 4)(2 d; 1 3). Nilai dari a + b + c + d adalah .....

Learn how to solve matrix equations involving matrix addition and multiplication to find the values of a, b, c, and d. This problem focuses on key matrix operations suitable for Grades 10-12 students, providing a detailed step-by-step approach.

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