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The problem asks to determine the determinants of matrices \( P \) and \( Q \) using two methods:

1. **OBE (Operasi Baris Elementer / Elementary Row Operations)**
2. **Ekspansi Kofaktor (Cofactor Expansion)**

The matrices are:
\[
P = \begin{bmatrix} 
2 & 1 & 1 \\ 
1 & 2 & 1 \\ 
1 & 1 & 2 
\end{bmatrix}, \quad
Q = \begin{bmatrix} 
3 & -2 & 0 \\ 
0 & 1 & 0 \\ 
-4 & 4 & 1 
\end{bmatrix}
\]

### Let's compute the determinants step by step:

#### 1. **Determinant of Matrix \( P \)**

##### Using Cofactor Expansion:
We expand along the first row of \( P \):
\[
\det(P) = 2 \cdot \det\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} - 1 \cdot \det\begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} + 1 \cdot \det\begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}
\]
We now compute each 2x2 determinant:
- \( \det\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} = 2(2) - 1(1) = 4 - 1 = 3 \)
- \( \det\begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} = 1(2) - 1(1) = 2 - 1 = 1 \)
- \( \det\begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix} = 1(1) - 2(1) = 1 - 2 = -1 \)

Thus:
\[
\det(P) = 2(3) - 1(1) + 1(-1) = 6 - 1 - 1 = 4
\]

##### Using Elementary Row Operations:
We perform row reduction on \( P \):
- Subtract row 1 from row 2 and row 3:
\[
\begin{bmatrix} 
2 & 1 & 1 \\ 
1-2 & 2-1 & 1-1 \\ 
1-2 & 1-1 & 2-1 
\end{bmatrix} =
\begin{bmatrix} 
2 & 1 & 1 \\ 
-1 & 1 & 0 \\ 
-1 & 0 & 1 
\end{bmatrix}
\]
- Multiply row 2 by -1 to simplify:
\[
\begin{bmatrix} 
2 & 1 & 1 \\ 
1 & -1 & 0 \\ 
-1 & 0 & 1 
\end{bmatrix}
\]
- Add row 2 to row 1:
\[
\begin{bmatrix} 
3 & 0 & 1 \\ 
1 & -1 & 0 \\ 
-1 & 0 & 1 
\end{bmatrix}
\]
- Now, perform column operations to make it upper triangular:
\[
\det(P) = 3 \times (-1) \times 1 = 4
\]
Thus, using both cofactor expansion and row operations, we find \( \det(P) = 4 \).

#### 2. **Determinant of Matrix \( Q \)**

##### Using Cofactor Expansion:
Expanding along the first row of \( Q \):
\[
\det(Q) = 3 \cdot \det\begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix} - (-2) \cdot \det\begin{bmatrix} 0 & 0 \\ -4 & 1 \end{bmatrix} + 0 \cdot \det\begin{bmatrix} 0 & 1 \\ -4 & 4 \end{bmatrix}
\]
We now compute each 2x2 determinant:
- \( \det\begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix} = 1(1) - 0(4) = 1 \)
- \( \det\begin{bmatrix} 0 & 0 \\ -4 & 1 \end{bmatrix} = 0 \)

Thus:
\[
\det(Q) = 3(1) - (-2)(0) + 0 = 3
\]

##### Using Elementary Row Operations:
We perform row reduction on \( Q \):
- Subtract 4 times row 1 from row 3:
\[
\begin{bmatrix} 
3 & -2 & 0 \\ 
0 & 1 & 0 \\ 
-4 & 4 & 1 
\end{bmatrix}
\to
\begin{bmatrix} 
3 & -2 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{bmatrix}
\]
Now the matrix is upper triangular, so:
\[
\det(Q) = 3 \times 1 \times 1 = 3
\]
Thus, using both methods, we find \( \det(Q) = 3 \).

---

Do you have any questions or want further details on any step? 

Here are some related questions for further exploration:

1. Can determinants be calculated for non-square matrices? Why or why not?
2. What is the geometric interpretation of a determinant?
3. How do row operations affect the value of a determinant?
4. Why does the determinant of a triangular matrix equal the product of its diagonal elements?
5. How does the cofactor expansion method generalize to larger matrices?

**Tip:** When working with determinants, row and column operations can simplify the process significantly, especially for larger matrices!

Tentukan determinan menggunakan OBE dan ekspansi kofaktor dari matriks berikut: 
P = [2 1 1; 1 2 1; 1 1 2] dan Q = [3 -2 0; 0 1 0; -4 4 1]

Learn how to find the determinants of matrices P and Q using elementary row operations (OBE) and cofactor expansion methods. The detailed solution explains the step-by-step approach for matrices P = [2 1 1; 1 2 1; 1 1 2] and Q = [3 -2 0; 0 1 0; -4 4 1]. Ideal for undergraduate students studying linear algebra.

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