F10HW7 - F2010 Homework 7 Section 3.4. 1.d) The...

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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 +...
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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.

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F10HW7 - F2010 Homework 7 Section 3.4. 1.d) The...

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