Let A be a square matrix such that A is not equal to I (Identity),A is not equal to -I with eigenvalue lambda.Prove that if A^2=I(I is the identity matrix),then lambda=1,lambda=-1
Let A be a square matrix such that A is not equal to I (Identity),A is...
[3] 7. Let A be a square matrix such that A# I and A+ -I with eigenvalue X. Prove that if AP = I(I is the identity matrix), then = 1 or = -1.
(3) 7. Let A be a square matrix such that A# I and A+ - with eigenvalue A. Prove that if AP = (is the identity matrix), then X = 1 or X = -1.
Let A be an ( n x n ) matrix, and let Lambda be an eigenvalue of A. Prove that for any scalar Alpha, Lambda + Alpha is an eigenvalue of A + Alpha x I (identity matrix).
In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in...
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is: ui l uentical...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
(i) Show that the following statements are equivalent for any square matrix A: Disg-. A is diagonalisable (i.e., A is similar to a diagonal matrix). Diag-2. R" has a basis of eigenvectors of A Diag-3. The algebraic and geometric multiplicity of each eigenvalue of A are equal.
6 8. Let A be a square matrix one of whose eigenvalues are 1. Is 12 an eigenvalue of B2. Why or why not?
2. Inverse of a square matrix: Determine the inverse matrix [A™'] of the given square matrix [A] using the Gauss-Jordan Elimination Method (GEM), and verify that [A-!] [A] = I where I is the identity matrix. A = [ 1 4 -27 0 -3 -2 | -3 4 1