Math Problem Statement
Find the inverse of the matrix.
Solution
The matrix provided is:
1 & 8 & 4 & 0 \\ 5 & 2 & 0 & 4 \\ -2 & 0 & 0 & -2 \\ 0 & 7 & 0 & -1 \end{bmatrix}$$ I will calculate the inverse of this matrix if it exists.import numpy as np # Define the given matrix matrix = np.array([[1, 8, 4, 0], [5, 2, 0, 4], [-2, 0, 0, -2], [0, 7, 0, -1]]) # Check if the matrix is invertible and compute its inverse if possible try: matrix_inverse = np.linalg.inv(matrix) except np.linalg.LinAlgError: matrix_inverse = "The matrix is not invertible." matrix_inversearray([[-4.19547438e-17, 1.40000000e+00, 3.00000000e+00, -4.00000000e-01], [ 1.46081977e-17, -2.00000000e-01, -5.00000000e-01, 2.00000000e-01], [ 2.50000000e-01, 5.00000000e-02, 2.50000000e-01, -3.00000000e-01], [ 4.51393309e-17, -1.40000000e+00, -3.50000000e+00, 4.00000000e-01]])array([[-4.19547438e-17, 1.40000000e+00, 3.00000000e+00, -4.00000000e-01], [ 1.46081977e-17, -2.00000000e-01, -5.00000000e-01, 2.00000000e-01], [ 2.50000000e-01, 5.00000000e-02, 2.50000000e-01, -3.00000000e-01], [ 4.51393309e-17, -1.40000000e+00, -3.50000000e+00, 4.00000000e-01]])successThe inverse of the given matrix is: $$\begin{bmatrix} -4.20 \times 10^{-17} & 1.4 & 3 & -0.4 \\ 1.46 \times 10^{-17} & -0.2 & -0.5 & 0.2 \\ 0.25 & 0.05 & 0.25 & -0.3 \\ 4.51 \times 10^{-17} & -1.4 & -3.5 & 0.4 \end{bmatrix}$$ Let me know if you want a detailed explanation of how this was calculated. --- Here are some related questions for deeper understanding: 1. What conditions must a square matrix satisfy to have an inverse? 2. How is the determinant of a matrix related to its invertibility? 3. What is the geometric interpretation of a matrix being invertible? 4. How can matrix inverses be applied in solving systems of linear equations? 5. Are there alternative methods (e.g., row reduction) to find a matrix inverse? **Tip:** For computational purposes, always check the determinant to verify a matrix is invertible before attempting to find its inverse.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Matrix Inversion
Formulas
Inverse of a matrix A is A^(-1) such that A * A^(-1) = I
Theorems
Invertible Matrix Theorem
Suitable Grade Level
Undergraduate Level