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The given problem asks for the calculation of the expression:

\[
3P^T - Q + 2R
\]

where:
- \( P = \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix} \)
- \( Q = \begin{pmatrix} 6 & -8 \\ 2 & 3 \end{pmatrix} \)
- \( R = \begin{pmatrix} -1 & -4 \\ 5 & 4 \end{pmatrix} \)
- \( P^T \) is the transpose of matrix \( P \).

### Steps to solve:
1. **Transpose of matrix \( P \):**
   \[
   P^T = \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix}
   \]

2. **Multiply \( P^T \) by 3:**
   \[
   3P^T = 3 \times \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 6 & -9 \\ 3 & 12 \end{pmatrix}
   \]

3. **Calculate \( 3P^T - Q \):**
   \[
   \begin{pmatrix} 6 & -9 \\ 3 & 12 \end{pmatrix} - \begin{pmatrix} 6 & -8 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 9 \end{pmatrix}
   \]

4. **Multiply matrix \( R \) by 2:**
   \[
   2R = 2 \times \begin{pmatrix} -1 & -4 \\ 5 & 4 \end{pmatrix} = \begin{pmatrix} -2 & -8 \\ 10 & 8 \end{pmatrix}
   \]

5. **Calculate the final result \( 3P^T - Q + 2R \):**
   \[
   \begin{pmatrix} 0 & -1 \\ 1 & 9 \end{pmatrix} + \begin{pmatrix} -2 & -8 \\ 10 & 8 \end{pmatrix} = \begin{pmatrix} -2 & -9 \\ 11 & 17 \end{pmatrix}
   \]

Thus, the final result is:

\[
\boxed{\begin{pmatrix} -2 & -9 \\ 11 & 17 \end{pmatrix}}
\]

Let me know if you need more details or have further questions.

### Related Questions:
1. What is the formula for matrix transposition?
2. How do you perform matrix multiplication by a scalar?
3. Can you explain matrix addition and subtraction?
4. What are some properties of matrix transposition?
5. How does matrix multiplication differ from element-wise multiplication?

### Tip:
Always ensure matrix dimensions match when performing addition and subtraction, as these operations are only defined for matrices of the same dimensions.

Diketahui matriks P = ( 2  1; -3  4 ), Q = ( 6  -8; 2  3 ), R = ( -1  -4; 5  4 ) dan PT adalah transpose dari matriks P, maka nilai dari 3PT - Q + 2R = ...

Learn how to solve the matrix expression 3P^T - Q + 2R with step-by-step calculations, including matrix transposition, scalar multiplication, and matrix addition and subtraction. Suitable for high school students in Grades 10-12.

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