Linear Algebra Answers

State De Moivre’s Theorem. (2)

9.2 Express cos 5θ and sin 4θ as polynomials in terms of sin θ and cos θ. (8)

9.3 Let w be a negative real number, z a 6

th root of w.

(a) Show that z (k) = ρ

1

6

-


cos

Consider the linear system

x + 2y + 3z = a

x + 3y + 8z = b

x + 2y + 2z = c







where a, b and c are arbitrary constants. Find all solutions of this system.

Consider the following homogeneous system of linear equations

x + 4y + z = 0

4x + 13y + 7z = 0

7x + 22y + 3z = 0







.

(a) Determine the solution(s) of the above system by reducing the system to row echelon

form. (7)

(b) Is (2, 4, 2) a solution of the system? Give a reason for your answer. (3)

(c) State how you can use the determinant of the coefficient matrix of the above system to determine

if the system has nontrivial solutions or not. DO NOT EVALUATE THE ACTUAL DETER-

MINANT.

Express the following as a linear combination of


p1 = 3 + x + 2x2, p2 = 5 - x + 5x2, and p3 = 2 + 5x + 3x2.

Let


1) a11 x1 + a12 x2 + a13 x3 = b1

2) a21 x1 + a22 x 2+ a23 x3 = b2

3) a31 x1 + a32 x 2+ a33 x 3 = b3


SHOW that if det(A) does not equal 0, where det (A) is the determinant of the coefficient matrix, then x2= det(A2)/det(A)


where det (A2) is the determinant obtained by replacing the second column of det (A) by (b1, b2, b3) to the power T.


Please assist.

LET A and B be any two matrices such that B in the inverse of A.


1) Determine the relationship between the adjoint of A and the adjoint of B (5 MARKS)


2) Determine the relationship between the transpose of A and the transpose of B (5 marks).


Please assist.

Solve the given matrix using strassen’s matrix multiplication

A matrix is 3 5

5 8

B matrix is 8 9

4 7

Find a c-inverse of the matrix

A D

2

4

1 0

2

Find the entries xi;i D 1; 2; 3; 4 in the matrix P so that P is an orthogonal matrix where

P D

1

2

2

6

6

4

1 1 1 1

1 1

[img]https://upload.cc/i1/2020/04/06/lcgHuW.jpg[/img]



question is in photo, R=4