Linear Algebra Answers

Define the determinant function and write its properties. Also use this definition
Find the determinant of a 3 × 3 matrix.

QUESTION 10

10.1 Use 9.2 to evaluate sin π/5 , sin 2π/5 and cos π/5 .

10.2 Let z = cos θ + i sin θ.

Then zn = cos(nθ) + i sin(nθ) for all n ∈ N (by de Moivre) and z−n = cos(nθ) − i sin(nθ).

(a) Show that 2 cos(nθ) = zn + z−n and 2i sin(nθ) = zn − z−n.
(2)

(b) Show that 2n cosn θ = (z + 1 )n and (2i)n sinn θ = (z − 1 )n.
(2)



(c) Use (b) to express sin7 θ in terms of multiple angles.
(6)

(d) Express cos3 θ sin4 θ in terms of multiple angles.
(6)

(e) Eliminate θ from the equations 4x = cos(3θ) + 3 cos θ; 4y = 3 sin θ − sin(3θ).
(5)

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Use Cramer's rule to find x in the equation systems below and stimulate your result by using any programming language.
Y-z=2
3x+2y+z=4
5x+4y=1

what is the difference between a singular and a non singular matrix,pls show that a non singular matrix must be square

Given the equation 4x-2y+6z=0
What values satisfy the equation when x=2 and z=1?
Define all elements of solution set in which the values of two variables equal zero.

consider the linear system
x+2y+3z=a
x+3y+8z=b
x+2y2z=c
where a,b and c are arbitrary constant.find all solutions of the system

Suppose

A= | 4 3 5 |
| 1 3 - 5 |
| 2 1 5 |.

1) Evaluate det(A) by expanding along the second row. No Other Method.

2) Can cramers rule be used to solve the system :

A | x |. | 0 |
| y | = | 0 |
| z | | a |

Where A is given and a can't equal 0?

Give reasons. IF YES, solve with cramers rule. IF NOT, solve with other method.

Please assist.

3. If the matrices A and B are given by A= and B =
a. Calculate 3A -2B+7I
b. 3A + 4X = B is an equation for an unknown matrix X. Find the matrix X
c. Calculate the matrix product AB
4. Solve the following set of equations by Gaussian elimination:
3x + 2y + z =2
4x + 2y + 2z =8
x – y + z = 8

Define T:R^3 -R^3 by
T(x1,x2,x3) = (x1+x2,x2+x3,x1-x3,2x1+x2-x3)
Check that T is a linear operator . Find the kernal and range of T. Find the dimension of the kernal.

Given:
A=
4 0 0
2 2 3
1 1 1
B=
4 0 0
2 2 3
1 1 1

Find: a) –A-1+ 3BT b) B-1+ ( AT+A-1)