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Exercise 7.5.13 Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample. (a) If A is a 3 x 3-matrix with determinant zero, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two nxn-matrices A and B, det(A + B) = det(A)+det(B).

Exercise 7.5.13 Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample. (a) If A is a 3 x 3-matrix with determinant zero, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two nxn-matrices A and B, det(A + B) = det(A)+det(B).

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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Exercise 7.5.13 Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample. (a) If A is a 3 x 3-matrix with determinant zero, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two nxn-matrices A and B, det(A + B) = det(A)+det(B). (d) For an n xn-matrix A, det (3A) = 3 det(A). (e) If A-¹ exists, then det(A-¹) = det(A)-¹. (f) If B is obtained by multiplying a single row of A by 4, then det(B) = 4 det(A). (g) For an nxn-matrix A, we have det(-A) = (-1)" det(A). (h) If A is a real nxn-matrix, then det(ATA) ≥ 0. (i) If Ak=0 for some positive integer k, then det(A) = 0. (j) If Ax=0 for some x 0, then det(A) = 0.
Transcribed Image Text:Exercise 7.5.13 Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample. (a) If A is a 3 x 3-matrix with determinant zero, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two nxn-matrices A and B, det(A + B) = det(A)+det(B). (d) For an n xn-matrix A, det (3A) = 3 det(A). (e) If A-¹ exists, then det(A-¹) = det(A)-¹. (f) If B is obtained by multiplying a single row of A by 4, then det(B) = 4 det(A). (g) For an nxn-matrix A, we have det(-A) = (-1)" det(A). (h) If A is a real nxn-matrix, then det(ATA) ≥ 0. (i) If Ak=0 for some positive integer k, then det(A) = 0. (j) If Ax=0 for some x 0, then det(A) = 0.
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