Math Problem Statement
Даны две матрицы A и B. Найти: a) AB; б) A^-1 и сделать проверку.
Solution
The task involves working with two matrices and . Here's what needs to be solved:
- Compute (the matrix product of and ).
- Compute (the inverse of ) and verify the result by confirming , where is the identity matrix.
Given matrices:
B = \begin{pmatrix} 1 & 4 & 4 \\ -2 & 2 & 4 \\ 5 & -3 & 5 \end{pmatrix}.$$ Let me calculate these step-by-step. numpy as np # Define matrices A and B A = np.array([[1, 2, 3], [-3, 1, -4], [-1, 4, 2]]) B = np.array([[1, 4, 4], [-2, 2, 4], [5, -3, 5]]) # Compute AB AB = np.dot(A, B) # Compute the inverse of A, if it exists A_inverse = np.linalg.inv(A) # Verify the inverse by computing A * A_inverse identity_check = np.dot(A, A_inverse) AB, A_inverse, identity_check### Solution: 1. **Matrix Product $$AB$$:** $$AB = \begin{pmatrix} 12 & -1 & 27 \\ -25 & 2 & -28 \\ 1 & -2 & 22 \end{pmatrix}$$ 2. **Inverse of Matrix $$A$$:** $$A^{-1} = \begin{pmatrix} 3.6 & 1.6 & -2.2 \\ 2.0 & 1.0 & -1.0 \\ -2.2 & -1.2 & 1.4 \end{pmatrix}$$ 3. **Verification of $$A^{-1}$$:** The product $$A \cdot A^{-1}$$ results in the identity matrix: $$I = \begin{pmatrix} 1 & 0 & -6.66 \times 10^{-16} \\ 0 & 1 & 0 \\ 8.88 \times 10^{-16} & 0 & 1 \end{pmatrix}.$$ The small numerical errors (close to $$0$$) are due to floating-point approximations in computation. Would you like further details or clarifications? ### Related Questions: 1. How do you determine if a matrix is invertible? 2. What are the properties of matrix multiplication? 3. Can the inverse of a matrix be computed for non-square matrices? 4. What methods are used to compute the inverse of a matrix by hand? 5. How is floating-point precision handled in numerical computations? ### Tip: To verify matrix inverses, always check if the product $$A \cdot A^{-1} = I$$ and $$A^{-1} \cdot A = I$$, where $$I$$ is the identity matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Inversion
Identity Matrix Verification
Formulas
Matrix product: AB = A * B
Matrix inverse: A * A^-1 = I
Theorems
Properties of matrix multiplication
Conditions for matrix invertibility
Suitable Grade Level
Undergraduate or Advanced High School
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