Let's analyze the given matrices $A$ and $B$.

Given matrices:

$A = \begin{pmatrix} 4a & 8 & 4 \\ 6 & -1 & -3b \\ 5 & 3c & 9 \end{pmatrix}$ $B = \begin{pmatrix} 12 & 8 & 4 \\ 6 & -1 & -3a \\ 5 & b & 9 \end{pmatrix}$

We are told that $A = B$. Therefore, we will compare each corresponding element of matrices $A$ and $B$.

Step-by-step comparison:

1. From the first row, first column: $4a = 12 \implies a = \frac{12}{4} = 3$

2. From the second row, third column: $-3b = -3a \implies -3b = -3(3) \implies -3b = -9 \implies b = 3$

3. From the third row, second column: $3c = b \implies 3c = 3 \implies c = \frac{3}{3} = 1$

Now, calculate $a + b + c$:

$a + b + c = 3 + 3 + 1 = 7$

Final Answer:

The value of $a + b + c$ is 7.

Would you like more details or have any questions?

Related Questions:

1. How do you determine if two matrices are equal?
2. What are some applications of matrices in real life?
3. Can you find other examples where matrix equality helps solve problems?
4. What are the properties of matrix addition and multiplication?
5. How do you solve systems of linear equations using matrices?

Tip:

When comparing matrices, always match each corresponding element to form equations for solving unknowns effectively.