Let's analyze the problem shown in the image.

We are given two matrices, $A$ and $B$, and we know that:

$A = \begin{pmatrix} 3a & 1 & 6 \\ 1 & 5 & -2b \\ 6 & 2c & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 12 & 1 & 6 \\ 1 & 5 & -2a \\ 6 & 6 & 4 \end{pmatrix}$

We are also given that $A = B$. Therefore, corresponding elements of both matrices must be equal.

By comparing corresponding elements of matrices $A$ and $B$:

1. In the first row, first column: $3a = 12 \quad \Rightarrow \quad a = \frac{12}{3} = 4$

2. In the second row, third column: $-2b = -2a \quad \Rightarrow \quad -2b = -2(4) \quad \Rightarrow \quad b = 4$

3. In the third row, second column: $2c = 6 \quad \Rightarrow \quad c = \frac{6}{2} = 3$

Now that we have the values of $a$, $b$, and $c$, the final step is to compute $a + b + c$:

$a + b + c = 4 + 4 + 3 = 11$

Thus, the answer is $a + b + c = 11$.

Do you need further clarifications? Here are 5 related questions to practice:

1. How would the solution change if the matrix $A$ had a different form?
2. What if the matrix $A$ was not equal to $B$, how would you solve for $a$, $b$, and $c$?
3. Can you generalize the method for solving for unknowns in two equal matrices?
4. How do you handle matrices of larger dimensions when comparing elements?
5. What are the properties of matrix equality?

Tip: When comparing matrices element-wise, ensure that both matrices have the same dimensions before proceeding with the comparison.