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)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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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 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98a85d3f-1221-4cd9-b5c2-c225eed9a46f%2Fc952cf7e-fcbc-4764-8408-69baf0042fe0%2Flrw76h_processed.jpeg&w=3840&q=75)
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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