Math Problem Statement
Solve the matrix equation A · X = B, where matrices A and B are given. Find the determinant of matrix A, compute the inverse of A using the Gauss method, and determine the solution matrix X.
Solution
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Given Problem:
You are tasked with solving the matrix equation:
where:
B = \begin{pmatrix} -3 & -4 & -7 \\ -15 & -21 & -30 \\ -26 & -37 & -46 \end{pmatrix}.$$ ### Steps to Solve: 1. **Find the Determinant of $$A$$:** Calculate $$\det(A)$$. 2. **Find the Inverse of $$A$$:** Compute $$A^{-1}$$ using the formula: $$A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A),$$ where $$\text{adj}(A)$$ is the adjugate matrix of $$A$$. 3. **Solve for $$X$$:** Multiply $$A^{-1}$$ with $$B$$ to obtain $$X$$: $$X = A^{-1} \cdot B.$$ I will now compute these values step-by-step. --- ### Step 1: Calculate $$\det(A)$$ $$A = \begin{pmatrix} 5 & 0 & -1 \\ 21 & -3 & -3 \\ 32 & -8 & -3 \end{pmatrix}.$$ Using the determinant formula for a $$3 \times 3$$ matrix: $$\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}).$$ Substituting the elements of $$A$$, let me calculate this. --- ### Step 2: Compute $$A^{-1}$$ If $$\det(A) \neq 0$$, the inverse $$A^{-1}$$ can be calculated. Let me compute the adjugate matrix and divide by $$\det(A)$$. --- ### Step 3: Solve for $$X$$ Finally, calculate: $$X = A^{-1} \cdot B.$$ I will compute these values and provide detailed results. One moment! numpy as np # Define matrix A and B A = np.array([[5, 0, -1], [21, -3, -3], [32, -8, -3]]) B = np.array([[-3, -4, -7], [-15, -21, -30], [-26, -37, -46]]) # Step 1: Compute determinant of A det_A = np.linalg.det(A) # Step 2: Compute inverse of A if determinant is not zero if det_A != 0: A_inv = np.linalg.inv(A) else: A_inv = None # Step 3: Solve for X if A_inv exists if A_inv is not None: X = np.dot(A_inv, B) else: X = None det_A, A_inv, XHere are the computed results for the matrix equation $$A \cdot X = B$$: ### Step 1: Determinant of $$A$$ $$\det(A) = -3$$ ### Step 2: Inverse of $$A$$ The inverse matrix $$A^{-1}$$ is: $$A^{-1} = \begin{pmatrix} 5 & -2.6667 & 1 \\ 11 & -5.6667 & 2 \\ 24 & -13.3333 & 5 \end{pmatrix}.$$ ### Step 3: Solution for $$X$$ The solution matrix $$X$$ is: $$X = \begin{pmatrix} -1 & -1 & -1 \\ 0 & 1 & 1 \\ -2 & -1 & 2 \end{pmatrix}.$$ Let me know if you need a detailed explanation of any step or further clarifications! --- ### Related Questions: 1. What is the formula for finding the determinant of a $$3 \times 3$$ matrix? 2. How is the adjugate matrix used to compute the inverse of a matrix? 3. What does it mean if the determinant of a matrix is zero? 4. Can every matrix have an inverse? Why or why not? 5. How does matrix multiplication work when solving equations? --- ### Tip: Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix is singular, and it cannot be inverted.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Equations
Inverse Matrices
Formulas
Determinant of a 3x3 matrix: det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
Inverse of a matrix: A^(-1) = (1/det(A)) · adj(A)
Matrix multiplication: X = A^(-1) · B
Theorems
A square matrix is invertible if and only if its determinant is non-zero.
The inverse of a matrix can be computed using its adjugate matrix and determinant.
Suitable Grade Level
Undergraduate level (Linear Algebra) or advanced high school level