Math Problem Statement

yes and use matrix inversion to detemine values

Solution

Using matrix inversion is an effective way to solve systems of linear equations. Let me explain the process and apply it with an example.

General Method:

For a system of linear equations: Ax=bA \mathbf{x} = \mathbf{b} Where:

  • AA is the coefficient matrix,
  • x\mathbf{x} is the vector of variables,
  • b\mathbf{b} is the constant vector.

The solution is given by: x=A1b\mathbf{x} = A^{-1} \mathbf{b} provided that AA is invertible (i.e., det(A)0\det(A) \neq 0).


Example:

Solve the system of equations using matrix inversion: 2x+y=52x + y = 5 4x3y=34x - 3y = 3

Step 1: Represent in matrix form

2 & 1 \\ 4 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$$ Here: $$A = \begin{bmatrix} 2 & 1 \\ 4 & -3 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$$ #### Step 2: Find $$A^{-1}$$ The inverse of a $$2 \times 2$$ matrix is calculated as: $$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \] Here: $$\det(A) = (2)(-3) - (1)(4) = -6 - 4 = -10$$ Thus: $$A^{-1} = \frac{1}{-10} \begin{bmatrix} -3 & -1 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} 0.3 & 0.1 \\ 0.4 & -0.2 \end{bmatrix}$$ #### Step 3: Multiply $$A^{-1}$$ by $$\mathbf{b}$$ $$\mathbf{x} = A^{-1} \mathbf{b} = \begin{bmatrix} 0.3 & 0.1 \\ 0.4 & -0.2 \end{bmatrix} \begin{bmatrix} 5 \\ 3 \end{bmatrix}$$ Compute: $$\begin{bmatrix} 0.3 \cdot 5 + 0.1 \cdot 3 \\ 0.4 \cdot 5 - 0.2 \cdot 3 \end{bmatrix} = \begin{bmatrix} 1.5 + 0.3 \\ 2.0 - 0.6 \end{bmatrix} = \begin{bmatrix} 1.8 \\ 1.4 \end{bmatrix}$$ #### Final Solution: $$x = 1.8, \quad y = 1.4$$ --- Would you like details on a specific step or need help with another example? --- ### Related Questions: 1. How can the determinant of a matrix determine its invertibility? 2. What are alternative methods to solve systems of equations besides matrix inversion? 3. Can you explain the difference between Cramer's Rule and matrix inversion? 4. How would the process change for a $$3 \times 3$$ system? 5. What is the geometric interpretation of matrix inversion? ### Tip: Always check the determinant before attempting to find the inverse of a matrix to ensure that it's not singular ($$\det(A) \neq 0$$).

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Systems of Linear Equations

Formulas

Matrix inversion formula: A^{-1} = (1/det(A)) * adj(A)
System solution: x = A^{-1} * b
Determinant for 2x2 matrix: det(A) = ad - bc

Theorems

Invertibility Theorem
Determinant Theorem

Suitable Grade Level

Grades 10-12