Find the eigenvalues of the matrix \( egin{bmatrix} 4 & 1 \ 2 & 3 \end{bmatrix} \).
["# How to Find the Eigenvalues of the Matrix ( \begin{bmatrix} 4 & 1 \ 2 & 3 \end{bmatrix} )", "Eigenvalues are fundamental concepts in linear algebra with wide applications in engineering, physics, computer science, and data analysis. Understanding how to compute the eigenvalues of a 2×2 matrix like\n[\nA = \begin{bmatrix} 4 & 1 \ 2 & 3 \end{bmatrix}\n]\nis essential for anyone working with eigenvalues. This article guides you step-by-step through the process of finding the eigenvalues of this matrix.", "---", "## What Are Eigenvalues?", "Eigenvalues are scalars ( \lambda ) such that for a square matrix ( A ), the non-zero vector ( \mathbf{v} ), called an eigenvector, satisfies the equation:", "[\nA\mathbf{v} = \lambda \mathbf{v}\n]", "Geometrically, eigenvalues represent scaling factors when the matrix ( A ) transforms a vector. The first step to finding eigenvalues is deriving a polynomial equation known as the characteristic equation.", "---", "## Step 1: Compute the Characteristic Equation", "The characteristic equation is derived from:", "[\n\det(A - \lambda I) = 0\n]", "where ( I ) is the identity matrix of the same size as ( A ). For our matrix:", "[\nA - \lambda I = \begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix}\n]", "The determinant of this matrix is:", "[\n\det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - (1)(2)\n]", "---", "## Step 2: Expand the Determinant", "Expand the expression:", "[\n(4 - \lambda)(3 - \lambda) = 12 - 4\lambda - 3\lambda + \lambda^2 = \lambda^2 - 7\lambda + 12\n]", "Now subtract 2:", "[\n\lambda^2 - 7\lambda + 12 - 2 = \lambda^2 - 7\lambda + 10\n]", "So, the characteristic equation is:", "[\n\lambda^2 - 7\lambda + 10 = 0\n]", "---", "## Step 3: Solve the Quadratic Equation", "Use the quadratic formula to solve for ( \lambda ):", "[\n\lambda = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 10}}{2 \cdot 1} = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm \sqrt{9}}{2} = \frac{7 \pm 3}{2}\n]", "This gives two solutions:", "[\n\lambda_1 = \frac{7 + 3}{2} = 5, \quad \lambda_2 = \frac{7 - 3}{2} = 2\n]", "---", "## Summary", "The eigenvalues of the matrix\n[\n\begin{bmatrix} 4 & 1 \ 2 & 3 \end{bmatrix}\n]\nare:", "[\n\boxed{2 \quad \ ext{and} \quad 5}\n]", "---", "## Why Are Eigenvalues Important?", "- Stability Analysis: In systems of differential equations, eigenvalues determine if solutions grow, decay, or oscillate.\n- Principal Component Analysis (PCA): In machine learning, eigenvalues help identify principal components and reduce data dimensionality.\n- Vibration Analysis: Found in mechanical and structural engineering to determine natural frequencies.", "---", "## Conclusion", "Finding eigenvalues of a 2×2 matrix involves forming and solving a characteristic polynomial from the determinant of ( A - \lambda I ). For the matrix ( \begin{bmatrix} 4 & 1 \ 2 & 3 \end{bmatrix} ), the eigenvalues are 2 and 5 — critical values that reveal key properties of the linear transformation defined by the matrix. Mastering this method empowers you to tackle more complex matrices and applications across science and engineering.", "---", "Keywords: Eigenvalues, eigenvector, characteristic equation, matrix diagonalization, linear algebra, mathematical methods, 2x2 matrix, quadratic equation, principal components.", "---", "If you want to practice, try finding eigenvalues of other 2×2 matrices using the same method — soon you’ll master this essential linear algebra skill!"]
