Linear Algebra Answers

Now suppose that A and B are arbitrary n  n matrices such that A and 2A + BT are invertible.

Using general properties of matrix operations show

that (A-1BT + 2I)-1 = (2A + BT)-1A. Show all the steps.

Given z=cos A + i sin A and u+iv =(1+z)(1+z^2).Prove that v=u tan (3A/2),u^2+ v^2=16 cos^2(A/2)cos^2(A).Hint:A=theta

consider the vectors;u(1,0)and v(0,1)

questions

1-determine cos theta,where theta is the angle between u and v.

2-determine the area of the parallelogram determined by u and v.



question 2

2.1-Let L1 and L2 be defined by x=W0+su where s is the element of real numbers

and y=w1+tv where t is the element of real numbers .

2.2-find the plane the passes through the point(2,4,-3)and is parallel to the plane -2x+4y-5z+5=0

2.3.find the line that passes through the point (2,5,3)and is perpendicular to the plane 2x-3y+4z+7=0

2.4find an equation of the plane passing through the point(-2,3,4)and is perpendicular to the line passing through the points (4,-2,5)and (0,2,4)

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.