Final Examination with Solution - Matrix Algebra | MATH 308, Exams of Mathematics

Download Final Examination with Solution - Matrix Algebra | MATH 308 and more Exams Mathematics in PDF only on Docsity! Final examination, solutions Math 308, Dec. 8th 2008 First Name: Last Name: Student ID: Section: The exam is 1h 50 minutes long. No textbook allowed. Students are allowed to bring two sheets of personal notes (two sided), and a scientific calculator. Exercise 1 Solve the following linear system of equations: x1 − x2 + x3 = 0 2x1 − x2 = 0 −2x1 + 3x2 − 3x3 = 10 The augmented matrix of the system is  1 −1 1 02 −1 0 0 −2 3 −3 10 . Its reduced echelon form is 1 0 0 100 1 0 20 0 0 1 10 , therefore there is a unique solution to the system: x1x2 x3  = 1020 10 . Exercise 2 Let A =  2 0 0 1 4 6 6 0 −2 −3 −3 0 0 0 0 2 . 1) Compute the characteristic polynomial of A and all its eigenvalues. [Hint: one of them is 3 and another is 2, check your computations if you did not find it] Let p(t) be the characteristic polynomial of A. p(t) = ∣∣∣∣∣∣∣∣ 2− t 0 0 1 4 6− t 6 0 −2 −3 −3− t 0 0 0 0 2− t ∣∣∣∣∣∣∣∣ = (2− t) ∣∣∣∣∣∣ 6− t 6 0 −3 −3− t 0 0 0 2− t ∣∣∣∣∣∣− ∣∣∣∣∣∣ 4 6− t 6 −2 −3 −3− t 0 0 0 ∣∣∣∣∣∣ = (2− t)2 ∣∣∣∣6− t 6−3 −3− t ∣∣∣∣− 0 = (2− t)2((6− t)(−3− t) + 18) = (t− 2)2(t− 3)t The eigenvalue of A are the roots of p(t), namely: • λ1 = 2 • λ2 = 3 • λ3 = 0 1 2) Compute the eigenspace for the eigenvalue λ = 3. Let E3 be the eigenspace for the eigenvalue λ2 = 3. E3 = N (A− 3I) = N   −1 0 0 1 4 3 6 0 −2 −3 −6 0 0 0 0 −1   . x ∈ N (A − 3I) if and only if x is solution of the homogeneous system (A − 3I)x = 0. The augmented matrix for this system reduces to the following reduced echelon form: −1 0 0 1 0 4 3 6 0 0 −2 −3 −6 0 0 0 0 0 −1 0  −→  1 0 0 0 0 0 1 2 0 0 0 0 0 1 0 0 0 0 0 0  . Thus x ∈ N (A− 3I) if and only if x1 = 0, x2 = −2x3, x4 = 0, and therefore E3 = Sp   0 −2 1 0   . 3) For the eigenvalue 2, determine its algebraic and its geometric multiplicity [Hint/reminder: com- pute the nullity of A− 2I]. The factor (t-2) appears at a power 2 in the characteristic polynomial of A, so the algebraic multiplicity of λ1 = 2 is 2. The geometric multiplicity of λ1 = 2 is the nullity of A − 2I, which in turn is the dimension of N (A− 2I). x ∈ N (A− 2I) iff  0 0 0 1 4 4 6 0 −2 −3 −5 0 0 0 0 0   x1 x2 x3 x4  = 0 The reduced echelon form of the augmented matrix for this homogeneous system is 1 0 −1/2 0 00 1 2 0 0 0 0 0 1 0 . Thus x ∈ N (A − 2I) iff x1 = 1/2x3, x2 = −2x3, x4 = 0 for x3 arbitrary. Finally N (A − 2I) = Sp   1/2 −2 1 0  . Its dimension is 1, and the geometric multiplicity of λ1 = 2 is 1. Exercise 3 Let V = R(A), (the range of the matrix A), where A =  1 2 3 2 1−1 −1 −2 2 2 2 −1 1 −1 2  1) Determine a basis for V and its dimension. Let B = AT =  1 −1 2 2 −1 −1 3 −2 1 2 2 −1 1 2 2 . Its reduced echelon form is  1 0 0 0 1 0 0 0 1 0 0 0 0 0 0  The non-zero rows of this matrix form a basis of the row-space of B. Therefore  10 0  , 01 0  , 00 1  is a basis for V = R(A). 2