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Suppose λ is an eigenvalue of matrix A and x is the corresponding eigenvector. Let x also be an eigenvector of the matrix B = A − 2I, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector x is equal to

1. λ
2. λ + 2
3. 2λ
4. λ − 2

Correct answer: λ − 2

Solution

If x is an eigenvector of A corresponding to eigenvalue λ, then Ax = λx. For the matrix B = A - 2I, we have Bx = (A - 2I)x = Ax - 2Ix = λx - 2x = (λ - 2)x. This shows that x is also an eigenvector of B with eigenvalue λ - 2.

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