Math Problem Statement
реши
реши
Solution
The given problem involves solving a matrix equation of the form:
1 & 2 & -3 \\ 3 & 2 & -4 \\ 2 & -1 & 0 \end{pmatrix} \cdot X = \begin{pmatrix} 1 & -3 & 0 \\ 10 & 2 & 7 \\ 10 & 7 & 8 \end{pmatrix}.$$ ### Solution Steps: 1. Let $$A$$ be the coefficient matrix: $$A = \begin{pmatrix} 1 & 2 & -3 \\ 3 & 2 & -4 \\ 2 & -1 & 0 \end{pmatrix}.$$ 2. Let $$B$$ be the right-hand side matrix: $$B = \begin{pmatrix} 1 & -3 & 0 \\ 10 & 2 & 7 \\ 10 & 7 & 8 \end{pmatrix}.$$ 3. Solve for $$X$$ using the equation: $$X = A^{-1} \cdot B,$$ where $$A^{-1}$$ is the inverse of $$A$$. ### Step 1: Compute the inverse of $$A$$. The formula for the inverse is: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A),$$ where: - $$\text{det}(A)$$ is the determinant of $$A$$, - $$\text{adj}(A)$$ is the adjugate of $$A$$. #### Compute $$\text{det}(A)$$: $$\text{det}(A) = 1 \begin{vmatrix} 2 & -4 \\ -1 & 0 \end{vmatrix} - 2 \begin{vmatrix} 3 & -4 \\ 2 & 0 \end{vmatrix} + (-3) \begin{vmatrix} 3 & 2 \\ 2 & -1 \end{vmatrix}.$$ Each minor determinant is: - $$\begin{vmatrix} 2 & -4 \\ -1 & 0 \end{vmatrix} = (2)(0) - (-4)(-1) = 0 - 4 = -4,$$ - $$\begin{vmatrix} 3 & -4 \\ 2 & 0 \end{vmatrix} = (3)(0) - (-4)(2) = 0 + 8 = 8,$$ - $$\begin{vmatrix} 3 & 2 \\ 2 & -1 \end{vmatrix} = (3)(-1) - (2)(2) = -3 - 4 = -7.$$ Substituting back: $$\text{det}(A) = 1(-4) - 2(8) + (-3)(-7) = -4 - 16 + 21 = 1.$$ #### Compute $$A^{-1}$$ (adjugate since $$\text{det}(A) = 1$$): Direct computation will be performed. ### Step 2: Multiply $$A^{-1}$$ with $$B$$. We’ll compute $$X = A^{-1} \cdot B$$ explicitly. Let me calculate this fully. numpy as np # Define the matrices A and B A = np.array([ [1, 2, -3], [3, 2, -4], [2, -1, 0] ]) B = np.array([ [1, -3, 0], [10, 2, 7], [10, 7, 8] ]) # Compute the inverse of A A_inv = np.linalg.inv(A) # Solve for X X = np.dot(A_inv, B) XThe solution to the given matrix equation is: $$X = \begin{pmatrix} 6 & 4 & 5 \\ 2 & 1 & 2 \\ 3 & 3 & 3 \end{pmatrix}.$$ ### Would you like additional details or further verification? Here are five related questions you might explore: 1. How is the determinant of a matrix computed, and why is it essential for finding the inverse? 2. What happens if the determinant of $$A$$ is zero? 3. Can this method be applied to non-square matrices? Why or why not? 4. How do we interpret the resulting matrix $$X$$ in terms of linear transformations? 5. What are some efficient computational techniques for solving large matrix equations? ### Tip: When solving matrix equations, always verify that the determinant is non-zero before attempting to compute the inverse!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Equations
Matrix Inversion
Formulas
Matrix equation: AX = B
Matrix inverse: A^-1
Determinant formula for 3x3 matrices
Theorems
Inverse of a matrix theorem
Determinant properties
Suitable Grade Level
College-level or Advanced High School (Grades 11-12)