MATH 240 Lecture Notes - Lecture 23: Polynomial Matrix, Diagonal Matrix, Invertible Matrix

• School: Simon Fraser University
• Department: Mathematics
• Course: MATH 240
• Professor: Michael Monagan

Document Summary

Please don"t distribute my solutions on course hero or anywhere on the internet. For the matrices a = (cid:20) 1 1. Calculate the eigenvalues of a and b, and, for each eigenvalue calculate a basis for the corresponding eigenvectors. Note, for b you should get eigenvalues 1, 3, 3. A i = (cid:20) 1 . The characteritics polynomial for a is det(a i) = (1 )(1 ) 1 = 2 2 = ( 2) So the eigenvalues of a are 0 and 2. 2 (cid:20) 1 t (cid:21) (cid:20) t t (cid:21) (cid:20) t. The characteristic polynomial for b is (1 )(3 )(1 ) + 2( 2)(3 ) = 3 5 2 3 + 9 = ( + 1) ( 3)2. So the eigenvalues of b are 1 and 3. Forming a i for each eigenvalue and row reducing it to rref, solving (a i)x = 0 for x to determine the eigenvectors, i get.