The larger of the two eigenvalues of the matrix \(\begin{bmatrix} 4 & 5 \\ 2 &…
2015
The larger of the two eigenvalues of the matrix \(\begin{bmatrix}
4 & 5 \\
2 & 1 \\
\end{bmatrix}
\) is _______.
Answer: 6 — Concept: An eigenvalue λ of a square matrix A satisfies det(A − λI) = 0. For a 2 × 2 matrix, the characteristic polynomial is λ² − (trace A)λ + det(A).…
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Correct answer: 6
Concept: An eigenvalue λ of a square matrix A satisfies det(A − λI) = 0. For a 2 × 2 matrix, the characteristic polynomial is λ² − (trace A)λ + det(A).
Application:
For A = [[4, 5], [2, 1]], trace A = 4 + 1 = 5 and det(A) = 4·1 − 5·2 = -6.
Therefore det(A − λI) = λ² − 5λ − 6 = 0.
Factor the polynomial: λ² − 5λ − 6 = (λ − 6)(λ + 1).
Hence λ = 6 or λ = -1, so the larger eigenvalue is 6.
Cross-check: The two values have sum 6 + (-1) = 5, equal to trace A, and product 6·(-1) = -6, equal to det(A). Both invariants agree with the matrix.
Result: 6