Linear Algebra Answers

a students is taking five course and is facing the crunch of final exams.she estimates that she has 40 hours available to study .if xj=the number of hours allocated to study for coursej.state the equation whose solution set specifies all possible allocation of time among the five courses which will exhaust the 40 hours

how that the set of all 3 x 3 upper
triangular matrices with entries from R is a
vector space over R under usual addition
and scalar multiplication of matrices. Find a
basis of this vector space.

Evaluate det (A) by expanding along the 2nd row.

Determine the area of the parallelogram determined by u(1,0) and v(0,1)

Solve the following system of linear equations by using the inverse matrix
method: (10)
x + y + z = 4
-2x - y + 3z = 1
y + 5z = 9

Find
a)-A raised to -1+3B raised to T
b)B raised to -1+(A raised to T+A raised to -1)

Let a11 x1 + a12 x2 + a13x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2
a31 x1 + a32 x2 + a33 x3 = b3.
Show that if det (A) 6= 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)T :
[

Determine the relationship between the adjoint of A and the adjoint of B.

Let z = cosθ + isinθ.
Then z^n = cos(nθ) + isin(nθ) for all n ∈ N(by de Moivre) and z^−n = cos(nθ)−isin(nθ).

(a) Show that 2cos(nθ) = z^n + z^−n and 2isin(nθ) = z^n −z^−n

(b) Show that 2^ncos^n θ = (z + 1/z)^n and (2i)^n sin^n θ = (z− 1 z/n)

(c) Use (b) to express sin^7 θ in terms of multiple angles

(d) Express cos^3 θsin^4 θ in terms of multiple angles

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

Let w be a negative real number, z a 6th root of w.

(a) Show that z (k) = ρ^1/6 [cos((π+2kπ)/6) + isin((π+2kπ)/6)], k = 0, 1, 2, 3, 4, 5 is a formula for the 6th roots of w. Show all your working.

(b) Hence determine the 6th roots of−729.

(c) Given z = cosθ + isinθ and u + iv = (1 + z)(1 + z^2). Prove that v = utan(3θ/2) and u2 + v2 = 16cos^2(θ/2)cos^2(θ)