Let's analyze the problem presented in the image.

The equation in question is:

$\begin{pmatrix} 5 & 7 \\ -1 & -5 \end{pmatrix} + 2 \begin{pmatrix} -1 & 4 \\ 2 & -2 \end{pmatrix} - 3Q = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

We are tasked with finding the sum of all elements in the matrix $Q$.

Step 1: Simplifying the equation

We begin by simplifying the expression on the left-hand side of the equation.

Multiply the second matrix by 2:

$2 \begin{pmatrix} -1 & 4 \\ 2 & -2 \end{pmatrix} = \begin{pmatrix} 2 \times (-1) & 2 \times 4 \\ 2 \times 2 & 2 \times (-2) \end{pmatrix} = \begin{pmatrix} -2 & 8 \\ 4 & -4 \end{pmatrix}$

Now the equation becomes: $\begin{pmatrix} 5 & 7 \\ -1 & -5 \end{pmatrix} + \begin{pmatrix} -2 & 8 \\ 4 & -4 \end{pmatrix} - 3Q = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

Step 2: Add the two matrices:

$\begin{pmatrix} 5 & 7 \\ -1 & -5 \end{pmatrix} + \begin{pmatrix} -2 & 8 \\ 4 & -4 \end{pmatrix} = \begin{pmatrix} 5 + (-2) & 7 + 8 \\ -1 + 4 & -5 + (-4) \end{pmatrix} = \begin{pmatrix} 3 & 15 \\ 3 & -9 \end{pmatrix}$

Now the equation becomes: $\begin{pmatrix} 3 & 15 \\ 3 & -9 \end{pmatrix} - 3Q = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

Step 3: Solve for $Q$

To isolate $Q$, move $3Q$ to the right-hand side: $\begin{pmatrix} 3 & 15 \\ 3 & -9 \end{pmatrix} = 3Q$

Now, divide both sides of the equation by 3 to solve for $Q$: $Q = \frac{1}{3} \begin{pmatrix} 3 & 15 \\ 3 & -9 \end{pmatrix} = \begin{pmatrix} 1 & 5 \\ 1 & -3 \end{pmatrix}$

Step 4: Find the sum of all elements in matrix $Q$

The sum of the elements in matrix $Q$ is: $1 + 5 + 1 + (-3) = 4$

Thus, the sum of all elements in matrix $Q$ is 4.

Final Answer:

The correct answer is B. 4.

Do you want any further details or explanations?

Here are 5 related questions for practice:

1. If $A + B - 2C = 0$, how would you solve for $C$?
2. What is the determinant of the matrix $\begin{pmatrix} 1 & 5 \\ 1 & -3 \end{pmatrix}$?
3. How do you perform scalar multiplication on a matrix?
4. What is the trace of a matrix, and how is it calculated?
5. How can matrix inversion be used to solve systems of linear equations?

Tip: When solving matrix equations, remember that scalar multiplication distributes across each element in the matrix.