Eigenvalues of the matrix $\begin{bmatrix} 5 & 3 \\ 1 & 4 \end{bmatrix}$ are
Options:
(A) -6.3 and -2.7
(B) -2.3 and -6.7
(C) 6.3 and 2.7
(D) 2.3 and 6.7
Official Answer Key:
D
Step-by-Step Solution & Explanation
Characteristic equation: det(A-λI)=0 → (5-λ)(4-λ)-3(1)=0 → λ²-9λ+17=0. Using the quadratic formula: λ = (9 ± √(81-68))/2 = (9 ± √13)/2 = (9 ± 3.606)/2. This gives λ₁ = 6.303 and λ₂ = 2.297, i.e. 2.3 and 6.3 → option (D). Option (C) is the same pair with a sign/order slip; (A) and (B) come from mis-signing the trace (9) or the discriminant.
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