Math Problem Statement
Solve the following matrix equation for X: [ [1, -1, 1], [2, 3, 0], [0, 2, -1] ] * X = [ [4, -1, 7, 9, 10], [6, 0, -5, 0, 1], [5, 7, -9, 4, 1] ]
Solution
The problem in the image is a matrix equation of the form:
1 & -1 & 1 \\ 2 & 3 & 0 \\ 0 & 2 & -1 \end{bmatrix} X = \begin{bmatrix} 4 & -1 & 7 & 9 & 10 \\ 6 & 0 & -5 & 0 & 1 \\ 5 & 7 & -9 & 4 & 1 \end{bmatrix}$$ You are asked to solve for the matrix $$X$$. ### Steps to Solve: This is a system of linear equations in matrix form $$AX = B$$, where $$A$$ is the 3x3 matrix on the left, and $$B$$ is the 3x5 matrix on the right. To solve for $$X$$, follow these steps: 1. **Multiply both sides by the inverse of $$A$$**: $$A^{-1} A X = A^{-1} B$$ Since $$A^{-1} A = I$$ (the identity matrix), this simplifies to: $$X = A^{-1} B$$ 2. **Find the inverse of matrix $$A$$**. 3. **Multiply the inverse of $$A$$ by matrix $$B$$** to get $$X$$. Let me calculate that for you.The matrix $$X$$ is: $$X = \begin{bmatrix} 21 & 18 & -1 & 39 & 32 \\ -12 & -12 & -1 & -26 & -21 \\ -29 & -31 & 7 & -56 & -43 \end{bmatrix}$$ This is the solution for the matrix equation. Let me know if you'd like more details, or if you have any questions! Here are 5 related questions you can explore: 1. How do you calculate the inverse of a matrix? 2. Can every matrix have an inverse? Why or why not? 3. What happens if a matrix is singular (non-invertible)? 4. How can matrix multiplication be used to solve systems of linear equations? 5. What are some real-life applications of matrix equations? **Tip**: Always check if a matrix is invertible by verifying its determinant is non-zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Equations
Matrix Inverse
Systems of Linear Equations
Formulas
Matrix equation: AX = B
Inverse of a matrix: A^{-1}
Solution: X = A^{-1} B
Theorems
Inverse Matrix Theorem
Determinant Theorem
Suitable Grade Level
Grades 10-12