Math Problem Statement
Find inverse of matrix 1,1,1,1. 1,2,4,8. 1,4,16,64. 1,5,25,125.
Solution
Let's find the inverse of the following matrix:
1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}$$ ### Step 1: Check if the matrix is invertible A matrix is invertible if its determinant is non-zero. We will first calculate the determinant of this matrix. For a $$4 \times 4$$ matrix, we can either compute the determinant directly or use a method like row reduction to simplify it. ### Step 2: Perform row reduction We will attempt to row reduce the matrix to row echelon form (REF) or reduced row echelon form (RREF). If we can reduce the matrix to the identity matrix, it will confirm that the matrix is invertible, and we can find the inverse. ### Step 3: Find the inverse Once the matrix is confirmed to be invertible, we can use the row reduction steps to find the inverse, or apply a formula for the inverse of a $$4 \times 4$$ matrix. Let me calculate the determinant and proceed with the steps for finding the inverse of this matrix. I'll begin by calculating the determinant of this matrix. numpy as np # Define the matrix A A = np.array([[1, 1, 1, 1], [1, 2, 4, 8], [1, 4, 16, 64], [1, 5, 25, 125]]) # Calculate the determinant of A det_A = np.linalg.det(A) # Check if the determinant is non-zero and calculate the inverse if so det_A, np.linalg.inv(A) if det_A != 0 else "Matrix is singular, no inverse."The determinant of the matrix is $$72$$, which is non-zero. This confirms that the matrix is invertible. The inverse of the matrix is: $$A^{-1} = \begin{bmatrix} 3.33 & -3.33 & 1.67 & -0.67 \\ -3.17 & 4.83 & -2.83 & 1.17 \\ 0.92 & -1.67 & 1.33 & -0.58 \\ -0.08 & 0.17 & -0.17 & 0.08 \end{bmatrix}$$ If you need further clarification or a breakdown of the process, feel free to ask! ### Follow-up questions: 1. How do I calculate the determinant of a $$4 \times 4$$ matrix step by step? 2. What are the properties of an invertible matrix? 3. How do I use Gaussian elimination to find the inverse of a matrix? 4. Can we apply these methods to matrices larger than $$4 \times 4$$? 5. How do we check if a matrix is singular (non-invertible)? ### Tip: When computing the determinant or inverse of larger matrices, breaking the problem down into smaller blocks (like row reduction or Laplace expansion) can help manage the complexity!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Determinants
Row Reduction
Formulas
Determinant formula for 4x4 matrices
Matrix inverse formula
Row reduction method for matrix inversion
Theorems
Invertible matrix theorem
Cramer's rule
Suitable Grade Level
Grades 11-12
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