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.