Q6: Define the following: nabla, unit vector, Matrix, Complex number, Differential equation? (5 Marks (5 Mark
4 Matrix A is defined as A = [3_21 (a) Find the eigenvalues. (5 marks) (b) Find a corresponding eigenvector for each of the eigenvalues found in (a). (10 marks) (c) Use the above (a) and (b) results to solve the vector-matrix differential equation * = 1} 21x with the initial conditions X(O) = (0) (10 marks)
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
Complex number has the form : z=a+bi, where a and b are real numbers and i is the imaginary unit(so, i=√-1. i2=-1). We define addition of complex numbers and scalar multiplication thusly:(a1+b1i)+(a2+b2i)=(a1+a2)+(b1+b2)ic(a+bi)=(ca)+(cb)ia) Show that the set "Z" of all complex numbers forms a vector space with the given operations.b) Find a Basis, the dimension of the vector space "Z" ,and the coordinates of z=3+4ic) Find the matrix, relative to the Basis you gave in Part (b), for the "conjugate" TransformationT:...
Inorganic Chemistry 1. Define the following terms: Ligand: Complex Coordination Number:
Solve the following equation in the complex number system. Please show all of your work. 2x^5-3x^4-32x+48=0
Question 5 a. (1 mark) Define a simple random sample of size n · (1 mark The following statement is True or False. Empirical Rule is applicable any data set (1 mark) A data set contains a data point that is extremely small, when compared to the rest of the data. The sample mean is (greater than)/ (less than) / (equal to) the c. sample median. d. (2 marks) For the following data set, 10, 16, 18, 22, 27 the...
Use the definition of Ax to write the vector equation as a matrix equation. 7 -5 -9 3 5 7 1 X1 + X2 +X3 -4 -4 - 7 7 -9 8 -3 4 X1 3 1 X2 7 X₂ (Type an integer or simplified fraction for each matrix element.)
(5) (10") Recall that for a unit vector ū in R, the matrix P = ūū represents the projection on ü. (a) Are there values a and b such that P is a SPD matrix? Explain. (b) Orthogonally diagonalize P. (c) Orthogonally diagonalize the reflection matrix L = 2P-I
F GHANA served) TEF RO HC 3 -2 0 A2. Given the matrix below 5 marks) [5 marks (10 marks (b) Compute explicitly the eigenvalues and determine the determinant, (c) Compute the corresponding eigenvectors of the matrix above (a) Show that the matrix is positive definite. 1 | so that the characteristic polynomial 5 marks 0 (d) Choose a, band c in the matrix B = | 0 Based on Cayley-Hamilton's theorem, every matrix fulfills its characteristic polynomial, using the...