Math Problem Statement
Dapatkan invers matriks berikut (3x3): [[1, 1, 1], [2, -1, 1], [3, -2, -1]]
Solution
The problem asks to find the inverse of the following matrix:
1 & 1 & 1 \\ 2 & -1 & 1 \\ 3 & -2 & -1 \end{pmatrix}$$ ### Steps to find the inverse of a $$3 \times 3$$ matrix: The inverse of a matrix $$A$$, if it exists, can be found using the formula: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$$ Where: - $$\text{det}(A)$$ is the determinant of matrix $$A$$, - $$\text{adj}(A)$$ is the adjugate (transpose of the cofactor matrix) of $$A$$. #### 1. Find the determinant of the matrix $$A$$: $$\text{det}(A) = 1 \left((-1)(-1) - 1(-2)\right) - 1 \left(2(-1) - 1(3)\right) + 1 \left(2(-2) - (-1)(3)\right)$$ Let me calculate the determinant first, and then proceed to find the inverse.The determinant of the matrix $$A$$ is $$\text{det}(A) = 7$$. Since the determinant is non-zero, the matrix is invertible. Now, we will proceed to find the adjugate matrix and calculate the inverse of $$A$$. Let me compute the inverse of the matrix next.The inverse of the matrix $$A$$ is: $$A^{-1} = \begin{pmatrix} 0.4286 & -0.1429 & 0.2857 \\ 0.7143 & -0.5714 & 0.1429 \\ -0.1429 & 0.7143 & -0.4286 \end{pmatrix}$$ If you need more details or have any questions, feel free to ask! Here are 5 related questions: 1. How is the adjugate of a matrix calculated? 2. Why is the determinant important in determining the invertibility of a matrix? 3. What happens if the determinant of a matrix is zero? 4. Can all matrices be inverted? Why or why not? 5. How can you use matrix inverses to solve systems of equations? **Tip**: The inverse of a matrix can also be used to solve linear systems using $$A^{-1} \cdot \mathbf{b} = \mathbf{x}$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Algebra
Matrix Inversion
Formulas
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Matrix Inversion Theorem
Determinant Theorem
Suitable Grade Level
Grades 11-12