Can you please solve a, b & c ? **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 \( 3 \times 3 \)-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 \( n \times n \)-matrices \( A \) and \( B \), \( \text{det}(A + B) = \text{det}(A) + \text{det}(B) \).(d) For an \( n \times n \)-matrix \( A \), \( \text{det}(3A) = 3 \text{det}(A) \).(e) If \( A^{-1} \) exists, then \( \text{det}(A^{-1}) = \text{det}(A)^{-1} \).(f) If \( B \) is obtained by multiplying a single row of \( A \) by 4, then \( \text{det}(B) = 4 \text{det}(A) \).(g) For an \( n \times n \)-matrix \( A \), we have \( \text{det}(-A) = (-1)^n \text{det}(A) \).(h) If \( A \) is a real \( n \times n \)-matrix, then \( \text{det}(A^T A) \geq 0 \).(i) If \( A^k = 0 \) for some positive integer \( k \), then \( \text{det}(A) = 0 \).(j) If \( Ax = 0 \) for some \( x \neq 0 \), then \( \text{det}(A) = 0 \).

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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)

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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