Math416_HW7_sol - Math 416 - Abstract Linear Algebra Fall...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Homework 7 solutions Section 3.3 3.5. Assume A is nilpotent, i.e. A k = for some k ≥ 1. Then we have det( A k ) = det( ) = 0 (det A ) k = 0 ⇒ det A = 0 . 3.6. Assume A and B are similar, i.e. A = SBS- 1 for some invertible matrix S . Then we have det A = det( SBS- 1 ) = (det S )(det B )(det S- 1 ) = (det S )(det B )(det S )- 1 = det B. 3.7. (2 pts) Assume Q is orthogonal, i.e. Q T Q = I . Then we have det( Q T Q ) = det I = 1 (det Q T )(det Q ) = 1 (det Q )(det Q ) = 1 ⇒ det Q = ± 1 . 1 3.8. (1 pt check) 1 x x 2 1 y y 2 1 z z 2 = 1 x x 2 y- x y 2- x 2 z- x z 2- x 2 = 1 x x 2 y- x ( y- x )( y + x ) z- x ( z- x )( z + x ) = ( y- x ) 1 x x 2 1 y + x z- x ( z- x )( z + x ) = ( y- x )( z- x ) 1 x x 2 0 1 y + x 0 1 z + x = ( y- x )( z- x ) 1 x x 2 0 1 y + x 0 0 z- y = ( y- x )( z- x )( z- y ) . Section 3.4 4.1. a. sign σ = sign(5 , 4 , 1 , 2 , 3) =- sign(1 , 4 , 5 , 2 , 3) = sign(1 , 2 , 5 , 4 , 3) =- sign(1 , 2 , 3 , 4 , 5) =- 1 . b. σ 2 = 1 2 3 4 5 5 4 1 2 3 ◦ 1 2 3 4 5 5 4 1 2 3 = 1 2 3 4 5 3 2 5 4 1 . c. σ- 1 = 1 2 3 4 5 5 4 1 2 3- 1 = 5 4 1 2 3 1 2 3 4 5 = 1 2 3 4 5 3 4 5 2 1 . 2 d. sign( σ- 1 ) = sign(3 , 4 , 5 , 2 , 1) =- sign(1 , 4 , 5 , 2 , 3) = sign(1 , 2 , 5 , 4 , 3) =- sign(1 , 2 , 3 , 4 , 5) =- 1 ....
View Full Document

This note was uploaded on 11/07/2011 for the course MATH 416 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

Page1 / 7

Math416_HW7_sol - Math 416 - Abstract Linear Algebra Fall...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online