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Mathematics 369 Homework (due Mar 22) A. Hulpke 39) For each of the following matrices, determine (with explanation) whether the matrix can be diagonalized. (You do not need to diagonalize it!) To save you calculations, the characteristic polynomial is given ineach case: 2 0 0 0 0 −16 −72 −316 0 97 3 2 950 0 −291 A = 48 222 , χA (x) = (x − 2) (x − 3) 10 36 147 3 −45 144 666 2844 0 −871 2 0 0 0 0 14 −3 −5 0 0 3 2 5 7 0 0 B= −14 , χB (x) = (x − 2) (x − 3) −28 10 8 3 0 0 0 0 0 3 341 −90 0 −48 30 1348 −356 0 −192 120 , χC (x) = (x − 1)(x − 2)(x − 3)(x − 4)(x − 5) −50 12 3 −4 7 C= −112 30 0 21 −10 46 −12 0 −4 6 40∗ ) The trace tr(A) of an n × n matrix A is the sum of its diagonal entries: tr(A) = a11 + a22 + · · · + ann . a) Show that for n × n matrices A, B we have that tr(AB) = tr(BA). (Hint: Use the formula ∑ j ai j b jk for matrix multiplication.) b) Show that similar matrices have the same trace. 41) The Fibonacci numbers Fn are defined by the recurrence F0 = F1 = 1, Fn = Fn−1 + Fn−2 (n ≥ 2). We want to determine a nonrecursive formula for the Fibonacci numbers. For this we will express them using matrix arithmetic. a) For n ≥ 1 let vn ∈ R2 be the column vector that contains the n-th and (n − 1)-th Fibonacci number: vn = (Fn , Fn−1 )T . (So v1 = (1, 1)T , v2 = (2, 1)T , v3 = (3, 2)T , v4 = (5, 3)T .) Show that 1 1 with the matrix A = we have that vn+1 = Avn . 1 0 b) Determine eigenvectors and eigenvalues for A. c) Find a diagonal matrix D and a matrix P such that D = P−1 AP. d) Determine a formula for An . e) Use the result of d) and the fact that vn+1 = An v1 to obtain a formula for Fn for general n. 42∗ ) Show that the characteristic xn − an−1 xn−1 − · · · − a2 x2 − a1 x − a0 . 0 1 0 A= 0 0 .. . 0 43) polynomial of the matrix A, given below, is χA (x) = 0 0 1 0 0 .. . 0 0 0 1 0 .. . 0 0 0 0 1 .. . ··· ··· ··· ··· ··· .. . ··· 0 0 0 0 0 0 a0 a1 a2 a3 a4 .. . 0 1 an−1 Let V = R3 with standard basis S and L:V → V defined by 2 −1 4 1 4 −3 S [L]S = 0 0 3 T a) Determine a basis h iB for the L-cyclic subspace W generated by the vector v = (0, 1, 0) . b) Determine L/W and the characteristic polynomial for L/W . B B c) Extend B to a basis C of V . h i d) Compute C [L]C . (Note that L/W is a submatrix!) B B e) Using C [L]C determine the characteristic polynomial for L. 44) Determine solutions to the following systems of linear differential equations for the given initial values: 3 −2 0 x(0) = (1, 1)T χA (x) = (x − 2)(x + 1) a) x (t) = · x(t), 2 −2 4 5 0 b) x (t) = x(0) = (0, 1)T χA (x) = x2 + 49 · x(t), −13 −4 46 −42 45) Let A = . Then χA (x) = (x − 4)(x − 25). Find four different matrices 21 −17 M1 , M2 , M3 , M4 ∈ R2×2 such that Mi2 = A. Problems marked with a ∗ are bonus problems for extra credit.