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The problem is asking to solve for \( r + s \) given the equation involving matrices:

\[
\begin{pmatrix}
\rho & q \\
2r & s
\end{pmatrix}
-
\begin{pmatrix}
2s & r \\
q & 2\rho
\end{pmatrix}
=
\begin{pmatrix}
2 & -2 \\
-2 & 2
\end{pmatrix}
\]

Let's break it down step by step:

### Step 1: Subtract the matrices
We subtract the elements of the two matrices on the left-hand side:

\[
\begin{pmatrix}
\rho - 2s & q - r \\
2r - q & s - 2\rho
\end{pmatrix}
=
\begin{pmatrix}
2 & -2 \\
-2 & 2
\end{pmatrix}
\]

### Step 2: Set up the system of equations
By comparing each corresponding element, we get the following system of equations:

1. \( \rho - 2s = 2 \)
2. \( q - r = -2 \)
3. \( 2r - q = -2 \)
4. \( s - 2\rho = 2 \)

### Step 3: Solve the system of equations

From equation (2) and (3), we solve for \( q \) and \( r \):
- From (2): \( q = r - 2 \)
- From (3): \( 2r - (r - 2) = -2 \)

Simplifying equation (3):
\[
2r - r + 2 = -2
\]
\[
r + 2 = -2
\]
\[
r = -4
\]

Now substitute \( r = -4 \) into equation (2):
\[
q = -4 - 2 = -6
\]

Now substitute \( r \) into equations (1) and (4):
1. \( \rho - 2s = 2 \)
2. \( s - 2\rho = 2 \)

From equation (1):
\[
\rho = 2s + 2
\]

Substitute \( \rho = 2s + 2 \) into equation (4):
\[
s - 2(2s + 2) = 2
\]
\[
s - 4s - 4 = 2
\]
\[
-3s - 4 = 2
\]
\[
-3s = 6
\]
\[
s = -2
\]

Now substitute \( s = -2 \) back into \( \rho = 2s + 2 \):
\[
\rho = 2(-2) + 2 = -4 + 2 = -2
\]

### Step 4: Find \( r + s \)
Since \( r = -4 \) and \( s = -2 \), we have:
\[
r + s = -4 + (-2) = -6
\]

Thus, \( r + s = -6 \).

---

Would you like further clarification on any of the steps? 

Here are some related questions to explore further:

1. What would happen if \( \rho \), \( q \), and \( s \) had different constraints?
2. How can matrix subtraction be used in solving systems of equations in general?
3. What role do matrix operations play in solving linear systems?
4. How can we generalize this problem to higher-dimensional matrices?
5. What are other methods to solve matrix equations besides comparison of elements?

**Tip**: When solving matrix equations, always ensure that each element of the matrix aligns correctly for operations like addition or subtraction.

Jika ρ, q, r, dan s memenuhi persamaan matriks maka ρ + q + r + s adalah ....

Solve a system of equations from matrix subtraction, where ρ, q, r, and s are variables. This problem focuses on matrix algebra and solving linear equations through comparison of matrix elements. Ideal for students in Grades 10-12.

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