A_PMT_2_M2J_F09 - Math 2J Lecture B Fall 2009 (Instructor:...

Info iconThis preview shows pages 1–2. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 2J Lecture B Fall 2009 (Instructor: Professor R. C. Reilly) Answer Key for Practice Midterm #2 (1) (a) Ill use the direct formula p ( ) =- 3 +( a 11 + a 22 + a 33 ) 2- ( A 11 + A 22 + A 33 ) +det( A ) that was discussed in class. By inspecting the entries of the matrix A = 4 2- 2 1 3- 1 1 1 1 , one sees that a 11 + a 22 + a 33 = 4 + 3 + 1 = 8; A 11 + A 22 + A 33 = (- 1) 1+1 3- 1 1 1 + (- 1) 2+2 4- 2 1 1 + (- 1) 3+3 4 2 1 3 = 4 + 6 + 10 = 20 . Finally, by using the cofactor expansion along the first row, one gets det( A ) = 4 A 11 + 2 A 12- 2 A 13 = 16- 4 + 4 = 16 . Thus, the characteristic polynomial is p ( ) =- 3 + 8 2- 20 + 16 . Remark The problem does not require you to factor this polynomial. If you did the factorization, however, and came up with eigenvalues other than the ones the problem tells you are the only ones, you should have figured out that you made a mistake! (b) The Case 1 = 2 One computes A- 2 I = 2 2- 2 1 1- 1 1 1- 1 2 2- 2 0 0 0 0 1 1- 1 0 0 0 0 The first step in the Gauss-Jordan process subtracts R 1 / 2 from R 2 and R 3 ; the second step divides R 1 by 2. The final matrix is in reduced row-echelon form, and it shows that the general solution x of the homogeneous system has two free variables, namely x 2 and x 3 . The formula is x = x 2 - 1 1 + x 3 1 1 Thus, the geometric multiplicity of the eigenvalue 1 = 2 for A is 2. The Case 2 = 4 Now one computes A- 4 I = 2- 2 1- 1- 1 1 1- 3 1- 1- 1 2- 2 1 1- 3 1- 1- 1 2- 2 2- 2 1- 1- 1 2- 2 1- 1- 1 1- 1 1 0- 2 0 1- 1 0 0 From this last matrix, which is in reduced row-echelon form, one reads off that the eigenspace for the eigenvalue 2 = 4 consists of all vectors x of the form x = x 3 2 1 1 In particular, the geometric multiplicity of 2 equals 1.equals 1....
View Full Document

Page1 / 4

A_PMT_2_M2J_F09 - Math 2J Lecture B Fall 2009 (Instructor:...

This preview shows document pages 1 - 2. Sign up to view the full document.

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