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Solved: Mark each statement True or False. Justify each
Chapter , Problem 1E(choose chapter or problem)
Mark each statement True or False. Justify each answer. Assume that all matrices here are square.a. If A is a 2 × 2 matrix with a zero determinant, then one column of A is a multiple of the other.b. If two rows of a 3 × 3 matrix A are the same, then det A = 0.c. If A is a 3 × 3 matrix, then det 5A = 5 det A.d. If A and B are n × n matrices, with det A = 2 and det B = 3, then det.(A + B) = 5.e. If A is n × n and det A = 2, then det A3 = 6.f. If B is produced by interchanging two rows of A, then det B = det A.g. If B is produced by multiplying row 3 of A by 5, then det B = 5 · det A.h. If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A. l. Any system of n linear equations in n variables can be solved by Cramer’s rule.m. If u and v are in then the area of the triangle in the plane with vertices at 0, u, and v is 10.
Questions & Answers
QUESTION:
Mark each statement True or False. Justify each answer. Assume that all matrices here are square.a. If A is a 2 × 2 matrix with a zero determinant, then one column of A is a multiple of the other.b. If two rows of a 3 × 3 matrix A are the same, then det A = 0.c. If A is a 3 × 3 matrix, then det 5A = 5 det A.d. If A and B are n × n matrices, with det A = 2 and det B = 3, then det.(A + B) = 5.e. If A is n × n and det A = 2, then det A3 = 6.f. If B is produced by interchanging two rows of A, then det B = det A.g. If B is produced by multiplying row 3 of A by 5, then det B = 5 · det A.h. If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A. l. Any system of n linear equations in n variables can be solved by Cramer’s rule.m. If u and v are in then the area of the triangle in the plane with vertices at 0, u, and v is 10.
ANSWER:Answer: Step-1: In this problem we need to mark each statement is true or false.1. Given statement: “ If A is a matrix with a zero determinant , then one column of A is a multiple of the other” is true. Because , if one column of A is a multiple of the other , then the columns of A are linearly dependent. Therefore , the given statement is true. b) Given statement: “ If two rows of a matrix A are the same , then detA = 0” is true. Because , If the two rows of matrix A are equal , then on subtraction of one row from the other , we will get a zero row , then the determinant of the matrix will become zero. Example: By , using row operation we get : det (A) = 0. Therefore , the given statement is true.Step-2:c) Given statement :” If A is a matrix , then det(5A) = 5 det(A)” is false. Because , if A is matrix , then Given A is a matrix . So , n = 3 Therefore , det(5A) = 125 det(A). Therefore , the given statement is false.d) Given statement : “If A and B are matrices , with det(A) = 2 and det(B) = 3 , then det(A+B) = 5” is false. Because , if the values of det(A) and det(B) are given , we cannot find the value of det(A+B) directly. Example: det(A) = 2 , det( B ) = 3 and det(A+B) = 12. . Therefore , the given statement is false.Step-3:e) Given statement : “ If A is and det(A) = 2 , then ” is