Discussion
Let A be a square matrix and let B = A - 0.2 I, where I is the identity matrix. Which of the following is always true ?
Answer: C
The statement that always holds is A and B have the same eigen vectors.
If \(Av = \lambda v\), then \(Bv = Av - 0.2v = (\lambda - 0.2)v\), and conversely. So every eigenvector of A is an eigenvector of B and vice versa; only the eigenvalues move, each one shifted down by 0.2.
That shift is exactly why 'same eigen values' fails. Rank can change: if 0.2 is an eigenvalue of A, then B is singular while A may be invertible (take A = 0.2 I, which has full rank, while B = 0 has rank 0). The same example shows that \(B^{-1}\) need not exist. So only the eigenvector statement is true for every square matrix A.
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