12. (1(a) You are given an n x n real anti-symmetric matrix A (i.e. A" = -A) and x € R".(i) ( s) What is the size of x" Ax?s) Prove that x" Ax = 0.- ) Determine whether there exists a 3 x 3 real symmetric matrix whose eigenvalues are(ii)(b) (гA1 = -1, Ag = 3, Az = 7and for which the corresponding eigenvectors are as follows:Vi =Va =V3If there is such a matrix, find it, and if there is none, explain why not.

Answered: Image /qna-images/answer/c40d6d4a-ee35-48c9-a011-98d05af57b12.jpg

12. (1 (a) You are given an n x n real anti-symmetric matrix A (i.e. A" = -A) and x € R". %3D +s) What is the size of x" Ax? s) Prove that x" Ax = 0. - ) Determine whether there exists a 3 x 3 real symmetric matrix whose eigenvalues are (i) ( (ii) (b) ( d1 = -1, A2 = 3, A3 = 7 and for which the corresponding eigenvectors are as follows: V2 = V3 If there is such a matrix, find it, and if there is none, explain why not.

12. (1 (a) You are given an n x n real anti-symmetric matrix A (i.e. A" = -A) and x € R". %3D +s) What is the size of x" Ax? s) Prove that x" Ax = 0. - ) Determine whether there exists a 3 x 3 real symmetric matrix whose eigenvalues are (i) ( (ii) (b) ( d1 = -1, A2 = 3, A3 = 7 and for which the corresponding eigenvectors are as follows: V2 = V3 If there is such a matrix, find it, and if there is none, explain why not.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

4: Let 6. A = | 0 7 0 0 7 (a) Find a matrix P that diagonalizes A. (b) Find PẬP [using your answer from (a.
-1 Suppose A, B, C, X are invertible n x n matrices. Solve the matrix equation C(B1X) = (B-²C-1) B² (A-³C)¯' for X, simplifying as much as possible (but no further!).
( 8.) Let A be a nonsingular n x n matrix, then there are elementary matrices G1, G2, such that Gg .. G2 G A =1. GR . True / False :
134. (Square roots of non-positive/normal matrices) First, show C about the similar matrix A = 01 0 G÷D 00 root, i.e. no matrix K such that K2 = C. Find all square roots of B = 10 000? 000 = 0 0 1 has no square 0 0 1 000 000 . How
i -1 2] Find the matrix A if (5A)" =|0 2 2 0 |3 -1 4
7. Prove that ||A|| = Omax (A) for a real matrix A.
|X= 3 -2 -2 2 ဦး + 0 1 + --L 1 Find a Basis for Column B of this Matrix
If you can answer my question quickly that would be amazing thank you!
:(" :) such that A? = C ) 13. Find all 2 x 2 matrices A =
A) Find the eigenbasis of the matrix and diagonalize it (if possible): when p=4. Also, Use this diagonalization to find A^3 and A^7. (the matrix in the photo) B) For the matrix in part (A), find A^2 and A^1 theorem.using Cayley-Hamilton