Linear Algebra Answers

If M,N,P are three matrices and M*N=I,and N *P=I where I is the identity matrix.Prove that M=P using associative law.

Find the point in R^3 in the plane x +2y -3z =12 closest to the point (1,1,1)

The advertising director for Karisma, a chain of four retail stores is considering 3 media possibilities: 1- Ads in the Ahram newspaper, 2- ads in a local UGO trade magazine that is
distributed free to all houses in the city and suburbs, and 3- ads on a
local Alex TV station. The stores are expanding their lines of Do -It-
Yourself tools and the advertising director is interested in a total new
customer exposure level of at least 50% within the city and 60% in the
suburbs. Each TV ad has a new-customer exposure level of 5% in the
city and 3% in the northwest suburbs. The newspaper ads have
corresponding exposure levels per ad of 3.5% and 3%, while the trade
magazine has corresponding exposure levels per ad of 0.5% and 1%.
The relevant costs are $1000 per newspaper ad, $300 per trade
magazine ad, and $2000 per TV ad.all 3 types of media
are used, Karizma would like to ensure that no single medium
consumes more than 45% of the total amount spent.(newspapers ads = 9, magazine ads = 30, TV = 1, Cost$20,000)

A retailer has two types of sugar priced at Rs 13 and Rs 18 per kg. He wants to sell the mixture of these 2 types at a uniform price of Rs 15 per kg. In what proportion should he mix them so that there is no change in his revenue? Hint: take x+y=100

Show that the set W = {(x 1 , x 2, x3, x4):
x1 + x2 + x3 + x4 = 0) is a subspace of R4.
Verify that (1, —1, 1, —1) and (1, 0, 0, —1)
are in W. Find a basis of W containing
(1,-1, 1, —1) and (1, 0, 0, —1).

A manufacture is to market a new fertilizer which is to be a mixture of ingredients A and B. The properties of the two ingredients are as give on the table below

properties of the ingredient Cost per kg
Bone meal Nitrogen Lime Phosphate
Ingredients A 20% 30% 40% 10% 12
B 40% 10% 50% 50% 8

It has been decided that the fertilizer will:

i. Be sold in bags containing 100 kilograms

ii. Be containing at least 15 kilograms of nitrogen

iii. Must contain at least 8 kilograms of phosphate

iv. Must contain at least 25 kilograms of bone meal

The manufacturer wants to meet the above requirements at minimum cost possible. Required
Develop a linear program and hence find the optimal solution for the problem.

Let W1 and W2 be subspaces of a vector space V and define
W1 + W2 := {u+v : u ∈ W1, v ∈ W2}.
Prove that span(W1 U W2) = W1 + W2.