2. Let A be a 2 x 2 matrix. Assume ₁ and ₂ are the two distinct non-zero eigenvalues of A. Determine whether the following statements are always true? If true, justify why. If not true, provide a couterexample. Statement A: If v₁ is an eigenvector corresponding to A₁ and v2 is an eigenvector corresponding to X2, then v₁ + v2 is an eigenvector of A, corresponding to eigenvalue X₁ + X₂. Statement B: If c E R, then cv₁ is an eigenvector of A, corresponding to X₁.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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2. Let A be a 2 × 2 matrix. Assume λ₁ and λ₂ are the two distinct non-zero eigenvalues of A. Determine
whether the following statements are always true? If true, justify why. If not true, provide a couterexample.
Statement A: If v₁ is an eigenvector corresponding to λ₁ and v2 is an eigenvector corresponding to №2,
then v₁ + v₂ is an eigenvector of A, corresponding to eigenvalue X₁ + X2.
Statement B: If c Є R, then cv₁ is an eigenvector of A, corresponding to X₁.
Transcribed Image Text:2. Let A be a 2 × 2 matrix. Assume λ₁ and λ₂ are the two distinct non-zero eigenvalues of A. Determine whether the following statements are always true? If true, justify why. If not true, provide a couterexample. Statement A: If v₁ is an eigenvector corresponding to λ₁ and v2 is an eigenvector corresponding to №2, then v₁ + v₂ is an eigenvector of A, corresponding to eigenvalue X₁ + X2. Statement B: If c Є R, then cv₁ is an eigenvector of A, corresponding to X₁.
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