Linear Algebra Answers

Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1 = {(1, 0, 0),(1, 1, 0),(0, 0, 1)},B2 = {(1, 0, 0),(0, 1, 2),(0, 2, 1)}. If Q(x) = x1^2-4x1x2+2x2x3+x2^2+x3^2 , find the representation of Q in terms of (y1,y2,y3)

a) Check whether the matrices A and B are diagonalisable. Diagonalise those
matrices which are diagonalisable.
A = 1 0 0 B = 2 0 0
1 5 -3 -2 2 -1
2 8 -5 -1 0 1
b) Find inverse of the matrix B in part a) of the question by using
Cayley-Hamiltion theorem.

c) Find the inverse of the matrix A in part a) of the question by fnding its
adjoint.

(b) If
A =


4 4 −2
−1 0 1
3 6 −1

 ,
show that A2 = A + kI for some constant k, where I is the unit matrix of order 3. Hence
find the inverse matrix A−1
. [

Popostino`s company has two pizza shops namely, Yesu Nkoaa and Yesu Nti at Assin Fosu and Winneba respectively. the number of Pacific Veggie, Pepperoni and Buffalo Chicken and large pizzas sold the last two weeks at at the two pizzerias is shown in the table below
Pizzerias Paccific Veggie Pepperoni Bufffalo Chicken
Yesu Nkoaa 350 504 413
Yesu Nti 248 622 512
The selling price for each pizzas is shown in the table below
Paccific Veggie Pepperoni Bufffalo Chicken
price $15.99 $23.99 $17.99

a. Write down a matrix multiplication that finds the total amount of income from sales of the three types of pizzas that each pizzeria generated during last two weeks.
b. Hence find the total amount of income from sales of the three types of pizzas that each pizzeria generated during last two weeks. Give your Answers correct to two decimal places.

(a) Let α : R
3 −→ R
3 be a linear transformation satisfying
α(1, 1, 0) = (1, 2, −1), α(1, 0, −1) = (0, 1, 1) and α(0, −1, 1) = (3, 3, 3).
i. Express (1, 0, 0) as a linear combination of (1, 2, −1), (0, 1, 1) and (3, 3, 3).

ii. Hence find v ∈ R
3
such that α(v) = (1, 0, 0).
(b) Let the map β : R
3 −→ R
3 be defined by
β((a, b, c)) = (a + b + c, −a − c, b)
for any (a, b, c) ∈ R
3
.
i. Show that β is a linear transformation.
ii. Find the kernel of β.

On a farmer’s market in Essex, the demand and supply functions for (a kilogram (kg) of) oranges (1), (a kg of) lemon (2) and (a kg of) nectarines (3) are:
q_1^d=-2p_1-4p_2+2p_3+160,q_1^s=2p_1+80
q_2^d=-4p_1-2p_2+5p_3+200,q_2^s=3p_2+160
q_3^d=2p_1+4p_2-5p_3+300,q_3^s=p_3+90
Use the equilibrium condition q_i^d=q_i^s to rewrite this system in the form Ap=d, where A is a 3×3 matrix of coefficients, p=(■(p_1@p_2@p_3 )), and d is a 3×1 vector. Then, use Cramer’s rule to solve for p_1 and p_3 (ONLY solve for p_1 and p_3!)

a) Let scalar be real numbers. Show that the set of real-valued, continuous functions on
the closed interval [0 ,1] forms a vector space.
b) Also show that those functions in part (a) for which all nth derivatives exist for

Prove that a set of n linearly independent vectors in a n–dimensional
vector space V span V

Qs: 04 Let T(x, y, z, w) = ( 3y -4z, x +2y+4z-w, 7z, -y-w)

Write the standard matrix for T. Check if T is one-to-one and onto.