\[ \det egin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix} = 0 \] - United Radiology

["Understanding the Characteristic Equation: Solving det\begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix} = 0", "When studying linear algebra and matrix theory, one fundamental task is finding the eigenvalues of a given square matrix. For the 2×2 matrix
\n[
\nA = \begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix},
\n]
\na key step is solving the determinant equation (\det(A - \lambda I) = 0), which leads to the characteristic polynomial. This article explores how to compute and interpret this equation, offering insight into eigenvalues and their significance.", "---", "### What is the Characteristic Equation?", "The characteristic equation arises from the expression (\det(A - \lambda I) = 0), where (I) is the identity matrix and (\lambda) represents an eigenvalue—a scaling factor that arises when a linear transformation (represented by matrix (A)) leaves a vector invariant up to scaling.", "For a (2\ imes2) matrix (A = \begin{bmatrix} a & b \ c & d \end{bmatrix}), the determinant-based characteristic equation simplifies to:
\n[
\n\lambda^2 - \ ext{tr}(A)\lambda + \det(A) = 0,
\n]
\nwhere (\ ext{tr}(A)) is the trace (sum of diagonal elements), and (\det(A)) is the determinant.", "---", "### Step 1: Compute the Matrix (A - \lambda I)", "Given:
\n[
\nA = \begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix}
\n]
\nSubtract (\lambda) times the identity matrix:
\n[
\nA - \lambda I = \begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix} - \begin{bmatrix} \lambda & 0 \ 0 & \lambda \end{bmatrix} = \begin{bmatrix} 4 - \lambda - \lambda & 1 \ 2 & 3 - \lambda - \lambda \end{bmatrix} = \begin{bmatrix} 4 - 2\lambda & 1 \ 2 & 3 - 2\lambda \end{bmatrix}
\n]", "---", "### Step 2: Compute the Determinant", "Compute (\det(A - \lambda I)):
\n[
\n\det\begin{bmatrix} 4 - 2\lambda & 1 \ 2 & 3 - 2\lambda \end{bmatrix} = (4 - 2\lambda)(3 - 2\lambda) - (1)(2)
\n]", "Expand the product:
\n[
\n(4 - 2\lambda)(3 - 2\lambda) = 12 - 8\lambda - 6\lambda + 4\lambda^2 = 4\lambda^2 - 14\lambda + 12
\n]", "Now subtract 2:
\n[
\n4\lambda^2 - 14\lambda + 12 - 2 = 4\lambda^2 - 14\lambda + 10
\n]", "So, the characteristic equation is:
\n[
\n4\lambda^2 - 14\lambda + 10 = 0
\n]", "---", "### Step 3: Simplify (Optional) and Solve the Quadratic", "Divide the entire equation by 2 to simplify:
\n[
\n2\lambda^2 - 7\lambda + 5 = 0
\n]", "Apply the quadratic formula:
\n[
\n\lambda = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2} = \frac{7 \pm \sqrt{49 - 40}}{4} = \frac{7 \pm \sqrt{9}}{4} = \frac{7 \pm 3}{4}
\n]", "Thus, the eigenvalues are:
\n[
\n\lambda_1 = \frac{10}{4} = 2.5, \quad \lambda_2 = \frac{4}{4} = 1
\n]", "---", "### Why Solve (\det(A - \lambda I) = 0)?", "- Finding eigenvalues: These values define vectors (eigenvectors) that remain directionally unchanged under transformation by (A).
\n- Matrix diagonalization: Eigenvalues are crucial in simplifying matrix powers and differential equations.
\n- System stability and dynamics: In applied fields like physics, control theory, and signal processing, eigenvalues determine system behavior (stability, oscillation, decay).", "---", "### Summary", "For the matrix
\n[
\n\begin{bmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{bmatrix},
\n]
\nsolving (\det(A - \lambda I) = 0) yields the characteristic quadratic, enabling computation of eigenvalues. Solving the simplified equation (2\lambda^2 - 7\lambda + 5 = 0) gives ( \lambda = 1 ) and ( \lambda = 2.5 )—critical values that unlock deeper understanding of linear transformations.", "---", "### Key Takeaways", "- Always subtract (\lambda I) from (A) to form (A - \lambda I).
\n- The determinant yields a quadratic equation whose roots are eigenvalues.
\n- This approach applies universally to (n \ imes n) matrices and forms the basis for advanced topics in linear algebra.", "For more insights into eigenvalues and quadrature computation, explore computational methods, numerical linear algebra tools, and applications across STEM fields.", "---", "Keywords: determinant equation, eigenvalue, characteristic polynomial, linear algebra, 2×2 matrix, eigenvectors, diagonalization, quadratic formula, linear transformation, mathematical solution, algebra eigenvalues, matrix analysis."]