For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification. (a) If A and B are symmetric 3 × 3 matrices with real entries, then 2A - 3B is a symmetric 3 × 3 matrix. (b) If A is a 5 × 3 matrix with real entries, then the rows of A must be linearly dependent. (c) If A and B are 2 × 2 matrices with real entries such that A² = B², then A = ±B. (d) If A is a 2 × 3 real-valued matrix with a zero column and B is any 3 × 3 real-valued matrix, then AB must have a zero column. (e) If vectors v, w = R³ are non-zero and orthogonal, then {v, w} is linearly independent.
For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification. (a) If A and B are symmetric 3 × 3 matrices with real entries, then 2A - 3B is a symmetric 3 × 3 matrix. (b) If A is a 5 × 3 matrix with real entries, then the rows of A must be linearly dependent. (c) If A and B are 2 × 2 matrices with real entries such that A² = B², then A = ±B. (d) If A is a 2 × 3 real-valued matrix with a zero column and B is any 3 × 3 real-valued matrix, then AB must have a zero column. (e) If vectors v, w = R³ are non-zero and orthogonal, then {v, w} is linearly independent.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.CR: Chapter Review
Problem 29E
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Transcribed Image Text:For each of the following statements, determine whether the statement is true or false. If the statement
is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false,
give an explicit numerical counterexample to the statement. No marks will be given without correct
justification.
(a) If A and B are symmetric 3 × 3 matrices with real entries, then 2A - 3B is a symmetric 3 × 3
matrix.
(b) If A is a 5 × 3 matrix with real entries, then the rows of A must be linearly dependent.
(c) If A and B are 2 × 2 matrices with real entries such that A² = B², then A = ±B.
(d) If A is a 2 × 3 real-valued matrix with a zero column and B is any 3 × 3 real-valued matrix, then
AB must have a zero column.
(e) If vectors v, w = R³ are non-zero and orthogonal, then {v, w} is linearly independent.
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