Suppose A and B are n x n invertible matrices, where n 1, and I is the n x n identity matrix. If A and B are similar matrices, which of the following statements must be true? I. A - 2I and B - 2I are similar matrices. II. A and B have the same trace. III. A^-1 and B^-1 are similar matrices.

Q: Suppose A and B are n x n invertible matrices, where n > 1, and I is the n x n identity matrix. If A and B are similar matrices, which of the following statements must be true? I. A - 2I and B - 2I are similar matrices. II. A and B have the same trace. III. A^-1 and B^-1 are similar matrices.

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $ A $ and $ B $ are similar matrices, there exists an invertible matrix $ P $ such that $ B = P^{-1}AP $. ‖ ‖ step 2 ⋮ To show $ A - 2I $ and $ B - 2I $ are similar, we can write $ B - 2I = P^{-1}AP - 2I $. ‖ ‖ step 3 ⋮ Using the property that $ P^{-1}IP = I $, we can rewrite $ B - 2I $ as $ P^{-1}AP - P^{-1}2IP $. ‖ ‖ step 4 ⋮ Simplifying, we get $ B - 2I = P^{-1}(A - 2I)P $, which shows that $ A - 2I $ and $ B - 2I $ are similar. ‖ ‖ step 5 ⋮ Since $ A $ and $ B $ are similar, they have the same trace, i.e., $ \text{trace}(A) = \text{trace}(B) $. ‖ ‖ step 6 ⋮ To show $ A^{-1} $ and $ B^{-1} $ are similar, we use the fact that $ B = P^{-1}AP $ to write $ B^{-1} = (P^{-1}AP)^{-1} $. ‖ ‖ step 7 ⋮ Applying the property of inverse of a product of matrices, we get $ B^{-1} = P^{-1}A^{-1}P $, which shows that $ A^{-1} $ and $ B^{-1} $ are similar. ‖ ∻Answer∻ ⚹ Statements I, II, and III must be true. ⚹ ∻Key Concept∻ ⚹Similar Matrices and Their Properties⚹ ∻Explanation∻ ⚹If two matrices $ A $ and $ B $ are similar, then $ A - \lambda I $ and $ B - \lambda I $ are also similar for any scalar $ \lambda $, they have the same trace, and their inverses $ A^{-1} $ and $ B^{-1} $ are similar as well.⚹◊What is the formula for the trace of a matrix A?⍭ Generate me a similar question◊

Question

Suppose A and B are n x n invertible matrices, where n > 1, and I is the n x n identity matrix. If A and B are similar matrices, which of the following statements must be true? I. A - 2I and B - 2I are similar matrices. II. A and B have the same trace. III. A^-1 and B^-1 are similar matrices.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $ A $ and $ B $ are similar matrices, there exists an invertible matrix $ P $ such that $ B = P^{-1}AP $. ‖ ‖ step 2 ⋮ To show $ A - 2I $ and $ B - 2I $ are similar, we can write $ B - 2I = P^{-1}AP - 2I $. ‖ ‖ step 3 ⋮ Using the property that $ P^{-1}IP = I $, we can rewrite $ B - 2I $ as $ P^{-1}AP - P^{-1}2IP $. ‖ ‖ step 4 ⋮ Simplifying, we get $ B - 2I = P^{-1}(A - 2I)P $, which shows that $ A - 2I $ and $ B - 2I $ are similar. ‖ ‖ step 5 ⋮ Since $ A $ and $ B $ are similar, they have the same trace, i.e., $ \text{trace}(A) = \text{trace}(B) $. ‖ ‖ step 6 ⋮ To show $ A^{-1} $ and $ B^{-1} $ are similar, we use the fact that $ B = P^{-1}AP $ to write $ B^{-1} = (P^{-1}AP)^{-1} $. ‖ ‖ step 7 ⋮ Applying the property of inverse of a product of matrices, we get $ B^{-1} = P^{-1}A^{-1}P $, which shows that $ A^{-1} $ and $ B^{-1} $ are similar. ‖ ∻Answer∻ ⚹ Statements I, II, and III must be true. ⚹ ∻Key Concept∻ ⚹Similar Matrices and Their Properties⚹ ∻Explanation∻ ⚹If two matrices $ A $ and $ B $ are similar, then $ A - \lambda I $ and $ B - \lambda I $ are also similar for any scalar $ \lambda $, they have the same trace, and their inverses $ A^{-1} $ and $ B^{-1} $ are similar as well.⚹◊What is the formula for the trace of a matrix A?⍭ Generate me a similar question◊

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