This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: F2010 Homework 7 Section 3.4. 1.d) The characteristic polynomial of the matrix is: 5 λ 1 4 1 2 λ 3 3 1 1 λ = (5 λ ) 2 λ 3 1 1 λ 1 3 3 1 λ + 4 1 2 λ 3 1 (5 λ )[(2 λ )(1 λ ) + 3] (1 λ 9) + 4[ 1 3(2 λ )] (5 λ )(2 3 λ + λ 2 + 3) + ( λ + 8) + 4(3 λ 7) = λ 3 + 8 λ 2 7 λ + 5 2.b) A 2 I =  1 1 1 1 1 2 3 3 4 ∼ 1 1 1 1 { x 1 + x 2 + x 3 = x 3 = The solution set of the homogeneous equation ( A 2 I ) x = is { x = b  1 1  b ∈ R } 3.e)f). e) The characteristic polynomial of the matrix is: 4 x 2 6 2 x 6 4 2 x = (4 x )(4 x 2 ) 2( 1)4(2 x ) = (2 x )[8 (4 x )(2 + x ) = x (2 x ) 2 The eigenvalues are 0 with multiplicity 1, and 2 with multiplicity 2. E : 4 2 6 2 6 ∼ 1 1 / 2 1 3 / 2 x 1 = c x 2 = 3 c x 3 = 2 c E = { x = c 1 3 2  c ∈ R } E 2 : 2 2 6 6 4 4 ∼ 1 1 x 1 = c x 2 = b x 3 = c E 2 = { x = c 1 1 + b 1 } . 1 f) 3 x 4 12 4 12 x 3 12 3 4 x = (3 x )[(12 + x )(4 + x ) 9] 4[4( 4 x ) 36]) + 12[12 + 12(12 + x )] = (3 x )[48 + 16 x + x 2 9] + 4(4 x + 4 × 13) + 12[12 + 12(12 +...
View
Full
Document
This note was uploaded on 12/05/2011 for the course M 341 taught by Professor Hietmann during the Spring '08 term at University of Texas at Austin.
 Spring '08
 HIETMANN

Click to edit the document details