Project ForgeEngineering Mathematics
Linear Algebra
GATE 2012
Medium
For the matrix A = [[5, 3], [1, 3]], ONE of the normalized eigenvectors is given as
- A(1/2, √3/2)
- (1/√2, −1/√2)
- C(3/√10, −1/√10)
- D(1/√5, 2/√5)
Explanation
Characteristic equation: λ²−(trace)λ+det = λ²−8λ+12=0, giving λ=6 or λ=2. For λ=2: (A−2I)v=0 → [[3,3],[1,1]]v=0 → v1+v2=0, so v=(1,−1), normalized to (1/√2,−1/√2). (Checking λ=6 gives eigenvector (3,1)/√10, which is NOT among the sign combinations offered as a correct match, confirming λ=2's eigenvector is the intended answer.)
Source: GATE ME 2012 (Booklet Code A) (2012)
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