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12. Let A be a square matrix of order n such that A² - I. Show that (i) the only eigenvalues of A are ±1; (ii) nullity(I + A) + nullity(I – A) = n; (iii) A is diagonalizable.

12. Let A be a square matrix of order n such that A² - I. Show that (i) the only eigenvalues of A are ±1; (ii) nullity(I + A) + nullity(I – A) = n; (iii) A is diagonalizable.

Advanced Engineering Mathematics
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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please do 12 i ii iii 
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10. A sequence {an} is defined by ao = a1 = az = 1 and an = 7an-2 + 6an-3 n2 3. Find a general formula for a, (in terms of n). (You may use MATLAB to find eigenvalues and eigenvectors.) 11. Let {u1, u2, . Un} be an orthogonal basis for R" and A = u,u + uzu+ un where the vectors are written as columns vectors. (i) Show that A is a symmetric matrix. (ii) Show that each of u1, Uz, ., Un is an eigenvector of A and find the corre- sponding eigenvalues. (iii) Find an orthogonal matrix P and diagonal matrix D such that PГАР- D. 12. Let A be a square matrix of order n such that A? = I. Show that (i) the only eigenvalues of A are +1; (ii) mullity(I + A) + nullity(I – A) = n; (ii) A is diagonalizable.
Transcribed Image Text:10. A sequence {an} is defined by ao = a1 = az = 1 and an = 7an-2 + 6an-3 n2 3. Find a general formula for a, (in terms of n). (You may use MATLAB to find eigenvalues and eigenvectors.) 11. Let {u1, u2, . Un} be an orthogonal basis for R" and A = u,u + uzu+ un where the vectors are written as columns vectors. (i) Show that A is a symmetric matrix. (ii) Show that each of u1, Uz, ., Un is an eigenvector of A and find the corre- sponding eigenvalues. (iii) Find an orthogonal matrix P and diagonal matrix D such that PГАР- D. 12. Let A be a square matrix of order n such that A? = I. Show that (i) the only eigenvalues of A are +1; (ii) mullity(I + A) + nullity(I – A) = n; (ii) A is diagonalizable.
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