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Official PYQ · 2015Set 2 · Q5Eigenvalues and Eigenvectors, LU DecompositionNAT1 mark

GATE CS 2015 Set 2 Q5 — Eigenvalues and Eigenvectors, LU Decomposition

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The larger of the two eigenvalues of the matrix [4521]\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix} is _______.
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Answer: 6

To find the eigenvalues of the matrix A=[4521]A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}, we use the characteristic equation:

\ndet⁡(A−λI)=0\det(A - \lambda I) = 0\n

For a 2×22 \times 2 matrix, this can be written as:

\nλ2−Trace(A)λ+det⁡(A)=0\lambda^2 - \text{Trace}(A)\lambda + \det(A) = 0\n

Let's calculate the trace and determinant of AA:

\n
    \n
  • Trace(A)=4+1=5\text{Trace}(A) = 4 + 1 = 5
    • \n
  • det⁡(A)=(4×1)−(5×2)=4−10=−6\det(A) = (4 \times 1) - (5 \times 2) = 4 - 10 = -6
    • \n
\n

Substituting these into the characteristic equation:

\nλ2−5λ−6=0\lambda^2 - 5\lambda - 6 = 0\n(λ−6)(λ+1)=0(\lambda - 6)(\lambda + 1) = 0\n

This gives the two eigenvalues:

\nλ1=6,λ2=−1\lambda_1 = 6, \quad \lambda_2 = -1\n

The larger of the two eigenvalues is 6.

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