Linear Algebra Answers

A company produces two products. Each product requires a certain amount of raw material. Product A requires 3 ponds of the raw material and product B 4 pounds. For any given week, the availability of the raw material is 2,400 pounds. If x equals the number of units produced of product A and Y the number of units of product B:
Determine the equation which states that total raw material used each week equals 2,400 pounds.
Rewrite the equation in slope-intercept form and identify the slope and Y-intercept.
Interpret the values of the slope and Y-intercept.

Show that C^n is a complex vector space

Write the application of Linear Algebra in Computer science and Engineering.

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!)