An electrical engineer is optimizing a power grid model and encounters a transformation matrix problem. Find the \(2 imes 2\) matrix \(\mathbf{M}\) such that for any vector \(\mathbf{v} = egin{bmatrix} x \ y \end{bmatrix}\), the transformation satisfies \(\mathbf{M}\mathbf{v} = egin{bmatrix} x + 2y \ 3x - y \end{bmatrix}\).

["Title: Find the Transformation Matrix (\mathbf{M}) That Maps (\mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix}) to (\begin{bmatrix} x + 2y \ 3x - y \end{bmatrix}) – A Step-by-Step Electrical Engineering Insight", "---", "In electrical engineering, modeling and transforming physical systems—such as power grid behaviors using matrix operations—is fundamental. A common challenge involves finding a linear transformation matrix (\mathbf{M}) that precisely maps input vectors to desired output forms. This article explores a classic transformation matrix problem: determining (\mathbf{M}) such that (\mathbf{M}\mathbf{v} = \begin{bmatrix} x + 2y \ 3x - y \end{bmatrix}), where (\mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix}).", "---", "### Understanding the Transformation", "We seek a (2 \ imes 2) matrix:", "[\n\mathbf{M} = \begin{bmatrix} a & b \ c & d \end{bmatrix}\n]", "such that when multiplying:", "[\n\mathbf{M}\mathbf{v} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} ax + by \ cx + dy \end{bmatrix}\n]", "this result must equal the target vector:", "[\n\begin{bmatrix} x + 2y \ 3x - y \end{bmatrix}\n]", "---", "### Setting Up the System of Equations", "Equating components gives two equations:", "1. (ax + by = x + 2y)\n2. (cx + dy = 3x - y)", "These must hold for all (x) and (y), so we match coefficients:", "From equation (1):\n- Coefficient of (x): (a = 1)\n- Coefficient of (y): (b = 2)", "From equation (2):\n- Coefficient of (x): (c = 3)\n- Coefficient of (y): (d = -1)", "---", "### Writing the Matrix (\mathbf{M})", "Substituting the coefficients:", "[\n\mathbf{M} = \begin{bmatrix} 1 & 2 \ 3 & -1 \end{bmatrix}\n]", "---", "### Verification", "Multiply (\mathbf{M}) with (\mathbf{v}):", "[\n\begin{bmatrix} 1 & 2 \ 3 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 \cdot x + 2 \cdot y \ 3 \cdot x + (-1) \cdot y \end{bmatrix} = \begin{bmatrix} x + 2y \ 3x - y \end{bmatrix}\n]", "The result matches the desired transformation exactly.", "---", "### Practical Application in Power Grid Modeling", "In smart power systems, such matrices represent state transformations—like converting operational parameters (voltage, current, phase angles) into model states for control and stability analysis. Optimal matrix design ensures efficient grid simulations, fault detection algorithms, and real-time load flow calculations. Understanding and computing these linear transformations empowers engineers to build more responsive and accurate models.", "---", "### Conclusion", "Finding the transformation matrix (\mathbf{M}) in matrix equation problems enables precise modeling of complex systems. For the transformation (\mathbf{v} \mapsto \begin{bmatrix} x + 2y \ 3x - y \end{bmatrix}), the required matrix is:", "[\n\mathbf{M} = \begin{bmatrix} 1 & 2 \ 3 & -1 \end{bmatrix}\n]", "This determinant ensures correct signal and power flow representations, essential in modern electrical engineering applications.", "---", "Optimizing power grid models starts with precise mathematics—matrix transformations are the backbone of advanced engineering analysis."]