2. (14pts) Diagonalize the following matrices, B = [ 4 1 1 1] 1 4 1...
1 0 4. Consider the matrices A = 0 +- Alw alcaldo and B o -1010 = 01. Answer the following o 0 2 questions. (5) Find all the vectors x and y which satisfy the following simultaneous equations. y = lim {A^ + B” k} n >00 \y\=1. Here, y is the length of the vector y.
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 a diagonal matrix D such that P-AP =D (VI) (2 marks) Find A107 by writing as linear combination of eigenvectors of A. (VII) (2 marks) Find a formula for Ak for all non-negative integers k. (Can k be a negative integer?) VIII) (1 mark) Use (VII)...
A= 10 5 5 2 B= 4. 1 > ,A+B= . ] 0
4-6. Using the Fourier transform integral, find Fourier transforms of the following signals: (a) xa(1)-1 exp(-α) u(t), α > 0; (b) xb(t) = u(t) u(1-t);
(1 point) Find the solution of x²y" + 5xy' + (4 – 3x) y = 0, x > 0 of the form y=x" Wazek, k=0 where ao = 1. r = help (numbers) ak = , k=1,2,3,... help (formulas)
Question 4 Find i(t) for t >0 for the circuit below. 4Ω 12 V 5 H 3 A
(4) (15 marks) Repeat the Question 2 for the following matrices -3 4 0] 0 0 A -2 30 B 0 -1 0 -8 8 1 0 0 1 ū= 10 = > 3 (I) (2 mark) Find the characteristic polynomial of matrix A. (II) (1 mark) Find eigenvalues of the matrix A. III) (2 mark) Find a basis for the eigenspaces of matrix A. IV) (1 mark) What is the algebraic and geometric multiplicities of its eigenvalues. (V) (2...
Prove or Disprove that:
If a > 0 and b are two rational numbers, then a' is a rational number.
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
Draw the remaining product of the following reaction Cl + 2 HC- CH2- NH2 > H3C-CH2-NH, Cl- +