True or False? Justify your answer. a) Let A, B, and C be two square matrices of the same dimension and A be an invertible matrix then ((A^−1)BCA)^2023 = A^(−1)xB^(2023)xC^(2023)xA. b) Let A be an n×n matrix. The set of all n×n matrices X that satisfies (A^2 −3I)X = O is not be closed under the matrix addition. (O denotes the zero matrix of the proper dimensions.) c) If A and B are 6×6 matrices of rank 6, then AB also has rank 6.
True or False? Justify your answer. a) Let A, B, and C be two square matrices of the same dimension and A be an invertible matrix then ((A^−1)BCA)^2023 = A^(−1)xB^(2023)xC^(2023)xA. b) Let A be an n×n matrix. The set of all n×n matrices X that satisfies (A^2 −3I)X = O is not be closed under the matrix addition. (O denotes the zero matrix of the proper dimensions.) c) If A and B are 6×6 matrices of rank 6, then AB also has rank 6.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.6: Matrices
Problem 9E
Related questions
Question
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True or False? Justify your answer.
a) Let A, B, and C be two square matrices of the same dimension and A be an invertible matrix then ((A^−1)BCA)^2023 = A^(−1)xB^(2023)xC^(2023)xA.
b) Let A be an n×n matrix. The set of all n×n matrices X that satisfies (A^2 −3I)X = O is not be closed under the matrix addition. (O denotes the zero matrix of the proper dimensions.)
c) If A and B are 6×6 matrices of rank 6, then AB also has rank 6.
Transcribed Image Text:Let A be an nxn matrix. The set of all n ×n matrices X that satisfies (A² −31)X = O
is not be closed under the matrix addition.
True
False
Transcribed Image Text:Let A, B, and C be two square matrices of the same dimension and A be an invertible
matrix then (A-¹BCA) 2023 = A-¹ B2023 C2023 A.
True
False
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