Linear Algebra Answers

1.) Determine the complex numbers i^2666 and i^145

2.) Let z1 =-i/-1+I and z2 =1+i/ 1- i. Express z1z3/z2, z1z2/z3, and z1/z3z2 in both polar and standard forms.

3.)Additional Exercises for practice:
Express z1 =-i, z2 =-1-i√3, and z3 = -√3 + i in polar form and use your results to find
(z4/3)
/z2/1 z -1/ 2 .
Find the roots of the polynomials below.
(a) P(z) = z2 + a for a > 0
(b) P(z) = z3-z2 + z-1.
(c) Find the roots of z3-1
(d) Find in standard forms, the cube roots of 8-8i
(e) Let w = 1 + i. Solve for the complex number z from the equation z^4 = w3.

4.)Find the value(s) for λ so that α = i is a root of P(z) = z^2 + λz-6.

1.
(a) Find a and b such that - 3ai-( - 1-i)b =3a-2bi.

(b) Let z1 = 12 + 5i and z2 = (3-2i)(2 + λi). Find λ without resorting to division such that z2 = z1.

2
Let z =2 + 3i and z^t = 5 - 4i. Determine the complex numbers
(a) z^2 -
zz^t
(b)1/2(z + z)^2
(c)1/2 [z-z] + [(- 1-z^t)]^2.

Find a and b such that -3ai-(-1 - i) b =3a - 2bi

Let z=z1/z2 where z1=tantheta +i and z2=z1. Find an expression for zn with n E N

Find the eigenvalues and an eigenvector per eigenvalue of the matrix
A= [ 0 1 1]
[ 3 -4 -3]
[ -5 7 6]
Is A is diagonalisable ? Give reason your answer.

i. Let A= [ 5 0 0]
[ 1 5 0]
[ 0 1 5]
Find a column vector X for which A= cX, for some c belong to R.

ii. Give an example with justification, of a skew-hermitian operator on C^2

Let v1= (1,0,-1), v2=(1,1,1) , v3= (2,2,3). Find the dual basis for {v1,v2,v3).

Find the inverse of the following matrix using row reduction method

[ 1 -1 2]
[ 3 2 4]
[ 0 1 -2]