Let A be an n x n matrix. • We define A to be happy if A² = A. • We define A to be melancholy if A² • We define A to be nihilistic if there is an integer k ≥ 1 such that Ak = On. (Recall that On is the n x n matrix of all zeroes.) = On. For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) Find a 2 x 2 melancholy matrix, not all of the entries of which are zero. (b) Find a matrix that is nihilistic but not melancholy. (c) Show that if an n x n matrix A is happy, then all of its eigenvalues are either 0 or 1. (d) True or false: If an n × n matrix A is nihilistic, then 0 is an eigenvalue of A. (e) True or false: If an n × n matrix A is happy, then both 0 and 1 are always eigenvalues of A.
Let A be an n x n matrix. • We define A to be happy if A² = A. • We define A to be melancholy if A² • We define A to be nihilistic if there is an integer k ≥ 1 such that Ak = On. (Recall that On is the n x n matrix of all zeroes.) = On. For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) Find a 2 x 2 melancholy matrix, not all of the entries of which are zero. (b) Find a matrix that is nihilistic but not melancholy. (c) Show that if an n x n matrix A is happy, then all of its eigenvalues are either 0 or 1. (d) True or false: If an n × n matrix A is nihilistic, then 0 is an eigenvalue of A. (e) True or false: If an n × n matrix A is happy, then both 0 and 1 are always eigenvalues of A.
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:Let A be an n × n matrix.
• We define A to be happy if A² = A.
• We define A to be melancholy if A² = On.
• We define A to be nihilistic if there is an integer k ≥ 1 such that Ak = On.
(Recall that On is the n x n matrix of all zeroes.)
For each part, determine if you believe the statement is true or false, and justify your answers as appropriate.
(a) Find a 2 × 2 melancholy matrix, not all of the entries of which are zero.
(b) Find a matrix that is nihilistic but not melancholy.
(c) Show that if an n x n matrix A is happy, then all of its eigenvalues are either 0 or 1.
(d) True or false: If an n × n matrix A is nihilistic, then 0 is an eigenvalue of A.
(e) True or false: If an n × n matrix A is happy, then both 0 and 1 are always eigenvalues of A.
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