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Are the following statements true or false? If true, explain why. If false, provide a counter example. (a) Consider the equation (r, y, 2) = (1, 1, 1) + (a, b, c)s + (a², &², c²)t for s,t € R. It is not possible to find 3 sets of values for a, b and e such that each set makes this equation describe a different geometric object. (b) It is possible to have two matricies A and B such that AB exists, BA exists, and yet A+ B does not exist. (c) The is no 2 x 2 matrix M such that M² = M unless M is the identity matrix or the zero matrix.

Are the following statements true or false? If true, explain why. If false, provide a counter example. (a) Consider the equation (r, y, 2) = (1, 1, 1) + (a, b, c)s + (a², &², c²)t for s,t € R. It is not possible to find 3 sets of values for a, b and e such that each set makes this equation describe a different geometric object. (b) It is possible to have two matricies A and B such that AB exists, BA exists, and yet A+ B does not exist. (c) The is no 2 x 2 matrix M such that M² = M unless M is the identity matrix or the zero matrix.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
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Are the following statements true or false? If true, explain why. If false, provide a counter example. (a) Consider the equation (r, y, 2) = (1, 1, 1) + (a, b, c)s + (a², &², c²)t for s,t € R. It is not possible to find 3 sets of values for a, b and e such that each set makes this equation describe a different geometric object. (b) It is possible to have two matricies A and B such that AB exists, BA exists, and yet A+ B does not exist. (c) The is no 2 x 2 matrix M such that M² = M unless M is the identity matrix or the zero matrix.
Transcribed Image Text:Are the following statements true or false? If true, explain why. If false, provide a counter example. (a) Consider the equation (r, y, 2) = (1, 1, 1) + (a, b, c)s + (a², &², c²)t for s,t € R. It is not possible to find 3 sets of values for a, b and e such that each set makes this equation describe a different geometric object. (b) It is possible to have two matricies A and B such that AB exists, BA exists, and yet A+ B does not exist. (c) The is no 2 x 2 matrix M such that M² = M unless M is the identity matrix or the zero matrix.
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