Math Problem Statement
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]
Solution
The problem asks to determine the determinants of matrices and using two methods:
- OBE (Operasi Baris Elementer / Elementary Row Operations)
- Ekspansi Kofaktor (Cofactor Expansion)
The matrices are:
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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Linear Algebra
Elementary Row Operations
Cofactor Expansion
Formulas
Determinant formula for 3x3 matrix
Cofactor expansion formula
Theorems
Properties of determinants
Cofactor expansion theorem
Elementary row operations properties
Suitable Grade Level
Undergraduate