Math Problem Statement
Can you help me solve 2 by 2 System of Equations with Matrices?
Solution
Yes, I can help you solve a system of equations using matrices! Here's a step-by-step explanation:
Problem Setup
A system of linear equations can be written as:
a_{11}x + a_{12}y &= b_1 \\ a_{21}x + a_{22}y &= b_2 \end{aligned}$$ This system can be expressed in matrix form as: $$\mathbf{A} \cdot \mathbf{x} = \mathbf{b}$$ where: $$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}.$$ To solve for $$\mathbf{x}$$, we use the formula: $$\mathbf{x} = \mathbf{A}^{-1} \cdot \mathbf{b}$$ ### Steps to Solve 1. **Determine $$\mathbf{A}$$:** Create the coefficient matrix from the system of equations. 2. **Find $$\mathbf{A}^{-1}$$:** Compute the inverse of the matrix $$\mathbf{A}$$, provided it is invertible. 3. **Compute $$\mathbf{x}$$:** Multiply the inverse of $$\mathbf{A}$$ by $$\mathbf{b}$$. --- ### Example Let's solve: $$\begin{aligned} 2x + 3y &= 8 \\ 4x - y &= 2 \end{aligned}$$ 1. **Write in matrix form:** $$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 8 \\ 2 \end{bmatrix}.$$ 2. **Find the determinant of $$\mathbf{A}$$:** $$\text{det}(\mathbf{A}) = (2)(-1) - (4)(3) = -2 - 12 = -14.$$ Since $$\text{det}(\mathbf{A}) \neq 0$$, $$\mathbf{A}$$ is invertible. 3. **Compute $$\mathbf{A}^{-1}$$:** The inverse is given by: $$\mathbf{A}^{-1} = \frac{1}{\text{det}(\mathbf{A})} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix}.$$ Substitute the values: $$\mathbf{A}^{-1} = \frac{1}{-14} \begin{bmatrix} -1 & -3 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} \frac{1}{14} & \frac{3}{14} \\ \frac{4}{14} & -\frac{2}{14} \end{bmatrix}.$$ 4. **Compute $$\mathbf{x} = \mathbf{A}^{-1} \cdot \mathbf{b}$$:** Multiply the matrices: $$\mathbf{x} = \begin{bmatrix} \frac{1}{14} & \frac{3}{14} \\ \frac{4}{14} & -\frac{2}{14} \end{bmatrix} \cdot \begin{bmatrix} 8 \\ 2 \end{bmatrix}.$$ Perform the multiplication: \[ \mathbf{x} = \begin{bmatrix} \frac{1}{Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
System of Linear Equations
Formulas
Matrix form: A * x = b
Inverse of a 2x2 matrix: A^(-1) = (1/det(A)) * [[a22, -a12], [-a21, a11]]
Solution: x = A^(-1) * b
Determinant of a 2x2 matrix: det(A) = a11*a22 - a12*a21
Theorems
Properties of Matrix Inverses
Determinant and Invertibility
Suitable Grade Level
Grades 10-12
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