7. You are given the following: 1 -14 4 ^-(3) ---) A = -1 6 -2 1-2 24 -7/ 4 a) Show that A has three distinct eigenvalues. b) Determine the corresponding eigenspaces. c) Show that the eigenspaces form a basis of R³.

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Which of the following statements is (are) true if A is an m x n matrix and V is a vector space (a) The dimension of the vector space P4 is 4 (b) The number of pivot columns of a matrix equals the deimension of its column space O (c) A vector space is infinite dimensional if it is span ned by an infinite set O (d) The nullity of A is the number of columns of A that are pivot columns O(e) dim Row A + nullity AT = m, the number of rows of A All of the above O None of the above O Options a, b, d and e O Options a, d and e O Options a, b and d O Options a, and e O Options b and e O Options b, d and e

Find the dimension of the eigenspace corresponding to the eigenvalue 1 = -3. -3 0 -3 1 0 -3

Consider the linear system X1 = and Find the eigenvalues and eigenvectors for the coefficient matrix. A₂ = 0 JI ← A, V₁ 9 V2 = 81 = = -3 6 - - 4 x. help (numbers) help (matrices) help (numbers) help (matrices) For each eigenpair in the previous part, form a solution of a' = Ax. Use t as the independent variable in your answers.

What are the Eigenvalues of the system below: X' 1 -(-2²) 1 O 2+i, 2-i O 1,2 O 1+2i, 1-2i O +2i, -2i X

Find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces. A = 4-1-1 = 1+i has eigenspace span has eigenspace span × !- (2-value with smaller imaginary part) (2-value with larger imaginary part)

For each matrix below, you are given one of the eigenvalues. Find a corresponding eigenvector for each eigenvalue: 3 1) A = 7 -2 -1 12 -7 and λ = 5. A corresponding eigenvector would be: 12 12 -10_ -8 + 8i 2) B= -12+9i 3) C = -3 + 15i 1+14i -48 +36i -12 +60i -40-25i -95-67i 53 +45i 204 +82i 37 + 15i 88 +45i -49-33i -180-50i 3-6i 3-6i and λ = 4-i. A corresponding eigenvector would be: 16-25i -129-9i 47+21i -317-61i 190 + 49i 616 - 32i 110 + 66i -63-45i -232-75i and λ = 1 + 2i. A corresponding eigenvector would be:

show that the i'ntegraL equation ges) -d S Ginssinet ) 96) 2+ 1 S(sins sinzt ) g(4) 2t has No eigen values.a

Verify that A, is an eigenvalue of A and that x, is a corresponding eigenvector. -1 4-6 4 5-12 A₁ = 13, x₁ = (1, 2, -1) A₂ = -3, x₂ = (-2,10) -2-4 3] A₂ = -3, x₂ = (3, 0, 1) A = -1 4 -6 4 5-12 -DE- I. -2-4 3 Ax₁ = -1 4 -6 -2 AND Ax₂ = 4 5-12 -2-4 3 -1 = 4 -6 Ax3 = 4 5-12 = 13 = -000- -2-4 3 1 41 2 = ₁x₁ -[] =-3 = 1 = ₂x₂ = 23x3 Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. (a) the characteristic equation P= (b) the eigenvalues (Enter your answers from smallest to largest.) (d₂d₂)= a basis for each of the corresponding eigenspaces x₁ = For the matrix A, find (if possible) a nonsingular matrix P such that P¹AP is diagonal. (If not possible, enter IMPOSSIBLE.) 2-2 7 03-2 0-1 -888- A = 2 Verify that P-¹AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP= 41