File Preview Assuming (AB)t = B¹A¹, det(AB) = det(A) det(B), and det (A¹) = det(A) complete any two of the following three. (a) If A is invertible, is the transpose of A, A¹, invertible? If so, what is (A¹)−¹? (b) If A is invertible, prove det (A−¹) = 1 det(A) (c) Suppose A = PBP-¹ for some invertible matrix P. Prove det(A) = det(B) and det(A - XI) = det(B − XI) where I is the identity matrix.
File Preview Assuming (AB)t = B¹A¹, det(AB) = det(A) det(B), and det (A¹) = det(A) complete any two of the following three. (a) If A is invertible, is the transpose of A, A¹, invertible? If so, what is (A¹)−¹? (b) If A is invertible, prove det (A−¹) = 1 det(A) (c) Suppose A = PBP-¹ for some invertible matrix P. Prove det(A) = det(B) and det(A - XI) = det(B − XI) where I is the identity matrix.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:File Preview Assuming (AB)t = B¹A¹, det(AB) = det(A) det(B), and det (A¹) = det(A)
complete any two of the following three.
(a) If A is invertible, is the transpose of A, A¹, invertible? If so, what is (Aª)−¹?
(b) If A is invertible, prove det (A−¹) =
1
det(A)
(c) Suppose A = PBP-¹ for some invertible matrix P. Prove
det(A) = det(B) and det(A - XI) = det(B − XI)
where I is the identity matrix.
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