Math Problem Statement
מנ המטריצה ההופכית של זה?
מנ המטריצה ההופכית של זה?
Solution
To calculate the inverse of the given matrix , we can proceed as follows:
Given Matrix
\begin{bmatrix} -2 & 11 & -4 \\ 1 & 3 & -1 \\ 0 & -3 & 1 \end{bmatrix}$$ The inverse of a matrix $$A$$, denoted $$A^{-1}$$, exists only if the determinant of $$A \neq 0$$. If it does exist, the inverse is calculated using: $$A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$$ where: - $$\text{det}(A)$$ is the determinant of $$A$$, - $$\text{adj}(A)$$ is the adjugate (transpose of the cofactor matrix) of $$A$$. Let me compute it step by step. numpy as np # Define the matrix P_C_to_B matrix = np.array([ [-2, 11, -4], [1, 3, -1], [0, -3, 1] ]) # Compute the determinant determinant = np.linalg.det(matrix) # Compute the inverse if determinant is non-zero inverse_matrix = None if determinant != 0: inverse_matrix = np.linalg.inv(matrix) determinant, inverse_matrixThe determinant of the matrix $$P_{C \to B}$$ is approximately $$1$$, meaning the matrix is invertible. The inverse of the matrix is: $$P_{C \to B}^{-1} = \begin{bmatrix} 0 & 1 & 1 \\ -1 & -2 & -6 \\ -3 & -6 & -17 \end{bmatrix}$$ ### Would you like to: 1. Verify this inverse by multiplying it with the original matrix? 2. Explore how the determinant was calculated? 3. Discuss the applications of matrix inverses in transformations? 4. See the derivation of the adjugate matrix for this case? 5. Learn about the numerical precision involved in matrix calculations? #### **Tip**: Always check the determinant before attempting to invert a matrix to ensure it’s non-zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Determinants
Adjugate Matrix
Formulas
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
det(A) ≠ 0 for matrix inversion
Theorems
Existence of the inverse matrix when determinant ≠ 0
Adjugate matrix and its relation to matrix inversion
Suitable Grade Level
Grades 10-12