Math Problem Statement
Represent the system of linear equations as a matrix (coefficient matrix, augmented matrix, and matrix equation AX = b).
Solution
The image contains a problem from Linear Algebra and Optimization, specifically focusing on Matrix Calculations. It asks to represent given systems of linear equations as:
- Coefficient Matrix (A)
- Augmented Matrix
- Matrix Equation (AX = b)
Here’s how you can break it down:
Example (a)
System of equations:
Coefficient Matrix (A):
\begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}$$ #### **Column Vector (b):** $$b = \begin{bmatrix} 8 \\ -2 \end{bmatrix}$$ #### **Augmented Matrix:** $$[A|b] = \begin{bmatrix} 2 & 3 & | 8 \\ 5 & -1 & | -2 \end{bmatrix}$$ #### **Matrix Equation:** $$AX = b$$ where $$X = \begin{bmatrix} x \\ y \end{bmatrix}$$ $$\begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ -2 \end{bmatrix}$$ Would you like me to do the same for the other systems (b, c, d)? ### **Related Questions:** 1. How do you find the determinant of a coefficient matrix? 2. What are the properties of an augmented matrix in Gaussian elimination? 3. How can you determine if a system has a unique solution, infinite solutions, or no solution? 4. What role does row echelon form play in solving systems of equations? 5. How do you compute the inverse of a matrix to solve AX = b? ### **Tip:** For a system of equations **AX = b**, if **det(A) ≠ 0**, the system has a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Calculations
Systems of Linear Equations
Formulas
Matrix Equation AX = b
Coefficient Matrix
Augmented Matrix
Theorems
Matrix Representation of Linear Systems
Suitable Grade Level
Grades 11-12