5. In this problem, you'll work with a new definition. Read it carefully before attempting the problems, and try to come up with some examples of matrices that satisfy and don't satisfy these properties. Definition 1 Let A be an n x n matrix. • We define A to be happy if A² = A. ● We define A to be glad if A² = In. (Recall that In is the n x n identity matrix.) True or false? For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) If A is non-zero and happy, then the homogeneous system Ar=0 has infinitely many solutions. (b) If A is glad, then the homogeneous system Aỡ = ♂ has only the trivial solution. (c) If A is happy, then In – A is happy.
5. In this problem, you'll work with a new definition. Read it carefully before attempting the problems, and try to come up with some examples of matrices that satisfy and don't satisfy these properties. Definition 1 Let A be an n x n matrix. • We define A to be happy if A² = A. ● We define A to be glad if A² = In. (Recall that In is the n x n identity matrix.) True or false? For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) If A is non-zero and happy, then the homogeneous system Ar=0 has infinitely many solutions. (b) If A is glad, then the homogeneous system Aỡ = ♂ has only the trivial solution. (c) If A is happy, then In – A is happy.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5. In this problem, you'll work with a new definition. Read it carefully before attempting the problems, and try
to come up with some examples of matrices that satisfy and don't satisfy these properties.
Definition 1
Let A be an n × n matrix.
• We define A to be happy if A² = A.
• We define A to be glad if A² = In. (Recall that In is the n × n identity matrix.)
True or false?
For each part, determine if you believe the statement is true or false, and justify your answers as appropriate.
(a) If A is non-zero and happy, then the homogeneous system Ar = 0 has infinitely many solutions.
(b) If A is glad, then the homogeneous system A = 0 has only the trivial solution.
(c) If A is happy, then In A is happy.
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