We are given the matrices $P$, $Q$, and $R$ and the equation:

$P + Q = 2R$

Where: $P = \begin{bmatrix} p & p-1 \\ p+2 & 6 \end{bmatrix}, \quad Q = \begin{bmatrix} 1 & -q \\ -7 & 2q \end{bmatrix}, \quad R = \begin{bmatrix} -2r & r \\ r & 7 \end{bmatrix}$

We need to determine the values of $p$, $q$, and $r$ such that this equation holds.

Step 1: Express $P + Q$

Adding $P$ and $Q$, we get:

$P + Q = \begin{bmatrix} p & p-1 \\ p+2 & 6 \end{bmatrix} + \begin{bmatrix} 1 & -q \\ -7 & 2q \end{bmatrix} = \begin{bmatrix} p+1 & p-1-q \\ p+2-7 & 6 + 2q \end{bmatrix}$ $P + Q = \begin{bmatrix} p+1 & p-1-q \\ p-5 & 6 + 2q \end{bmatrix}$

Step 2: Express $2R$

Multiplying $R$ by 2:

$2R = 2 \times \begin{bmatrix} -2r & r \\ r & 7 \end{bmatrix} = \begin{bmatrix} -4r & 2r \\ 2r & 14 \end{bmatrix}$

Step 3: Set up the system of equations

From the equation $P + Q = 2R$, we equate corresponding elements from the matrices:

1. $p+1 = -4r$
2. $p-1-q = 2r$
3. $p-5 = 2r$
4. $6 + 2q = 14$

Step 4: Solve the system

From equation (4): $6 + 2q = 14 \quad \Rightarrow \quad 2q = 8 \quad \Rightarrow \quad q = 4$

From equation (3): $p - 5 = 2r \quad \Rightarrow \quad p = 2r + 5$

Substitute $p = 2r + 5$ into equation (1): $2r + 5 + 1 = -4r \quad \Rightarrow \quad 2r + 6 = -4r \quad \Rightarrow \quad 6r = -6 \quad \Rightarrow \quad r = -1$

Now substitute $r = -1$ into $p = 2r + 5$: $p = 2(-1) + 5 = 3$

Final values:

$p = 3, \quad q = 4, \quad r = -1$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What are the properties of matrices that allow us to perform element-wise operations like addition?
2. Can this method be extended to solve for 3x3 matrices or higher dimensions?
3. What are the implications of scalar multiplication on matrix transformations?
4. How would the problem change if $P + Q = 3R$ instead of $2R$?
5. Could we use determinant properties to check the validity of the solution?

Tip: When solving matrix equations, always make sure to compare corresponding elements carefully to avoid mistakes.