Two n × n square matrices A and B are said to be similar if there exists a non-singular matrix P such that P–1A, P = B. If A and B are similar matrices such that det(A) = 1, then

Engineering

Mathematics

Introduction to Determinants

Question

Two n × n square matrices A and B are said to be similar if there exists a non-singular matrix P such that P^–1A, P = B. If A and B are similar matrices such that det(A) = 1, then

det(B) = 1

det(A) = – det(B)

det(B) = – 1

none of these

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Given, two n × n square matrices A and B

Given, B = P^–¹AP

det⁡(B) = det⁡(P^–¹AP)

det⁡(B) = (det(⁡P^–¹)) (det⁡(A)) (det⁡(P))

det⁡(B) = (det⁡(P))^–¹(det⁡(A)) (det⁡(P)) [∵ | A | | A^–¹ | = AA^–¹ | = | I | = 1]

det⁡(B) = det⁡(A) = 1

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