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In Exercises 27 and 28, A and B are n × n matrices. Mark
Chapter 3, Problem 27E(choose chapter or problem)
In Exercises 27 and 28, A and B are n × n matrices. Mark each statement True or False. Justify each answer.a. A row replacement operation does not affect the determinant of a matrix.b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by where r is the number of row interchanges made during row reduction from A to U.c. If the columns of A are linearly dependent, then det A = 0.
Questions & Answers
QUESTION:
In Exercises 27 and 28, A and B are n × n matrices. Mark each statement True or False. Justify each answer.a. A row replacement operation does not affect the determinant of a matrix.b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by where r is the number of row interchanges made during row reduction from A to U.c. If the columns of A are linearly dependent, then det A = 0.
ANSWER:Solution 27E(a)Use the following result to decide that the given statement is whether true or false.Let A be a n x n matrix and E be an n x n elementary matrix, then det (EA) = det (E) det (A)If is a row replacement, then det (E) = 1.det (EA) = det (E) det (A) = (1) det (A) = det (A)Thus, the row replacement does not affect on the determinant of the matrix.Hence, the given statement is true.(b)Suppose the n x n matrix, A has been reduced to an echelon form U by row replacements and row operations and the number of interchanges is r.If two rows of A are interchanged to produce B, then det B= -det AIf there are r number of times pair of rows interchanges in A, then det