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(a) State the three types of elementary row operations on m x n matrices, where m, n > 1. (b) Regarded in M3 (F5), where F5 is the field of five elements, put the matrix [024] 420 0 3 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4−¹ = 4 when working in F5.) (c) the determinant of a matrix changes under each type of elementary row operation. Use this to deduce det(A) for the matrix in part (b). (d) Let A be any square matrix over a field K. Is A necessarily equivalent to At, the transpose of A? Justify your answer. (Hint: you may assume usual facts about the transpose, including that if P is invertible then (Pt)-¹ = (P-¹) and similarly for Q.)

(a) State the three types of elementary row operations on m x n matrices, where m, n > 1. (b) Regarded in M3 (F5), where F5 is the field of five elements, put the matrix [024] 420 0 3 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4−¹ = 4 when working in F5.) (c) the determinant of a matrix changes under each type of elementary row operation. Use this to deduce det(A) for the matrix in part (b). (d) Let A be any square matrix over a field K. Is A necessarily equivalent to At, the transpose of A? Justify your answer. (Hint: you may assume usual facts about the transpose, including that if P is invertible then (Pt)-¹ = (P-¹) and similarly for Q.)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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(a) State the three types of elementary row operations on m × n matrices, where m, n > 1. (b) Regarded in M3(F5), where F5 is the field of five elements, put the matrix 024 4 20 003 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4-1 = 4 when working in F5.) the determinant of a matrix changes under each type of elementary row operation. Use this to deduce det(A) for the matrix in part (b). (c) (d) Let A be any square matrix over a field K. Is A necessarily equivalent to At, the transpose of A? Justify your answer. (Hint: you may assume usual facts about the transpose, including that if P is invertible then (Pt)-¹ = (P-¹) and similarly for Q.)
Transcribed Image Text:(a) State the three types of elementary row operations on m × n matrices, where m, n > 1. (b) Regarded in M3(F5), where F5 is the field of five elements, put the matrix 024 4 20 003 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4-1 = 4 when working in F5.) the determinant of a matrix changes under each type of elementary row operation. Use this to deduce det(A) for the matrix in part (b). (c) (d) Let A be any square matrix over a field K. Is A necessarily equivalent to At, the transpose of A? Justify your answer. (Hint: you may assume usual facts about the transpose, including that if P is invertible then (Pt)-¹ = (P-¹) and similarly for Q.)
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