Linear Algebra Answers

Prices to a concert cost N$200 for an adult and N$150 for a child, the concert venue can
accommodate at most 240 people. The organizers are giving a N$20 discount to every adult
and N$10 discount for every child but do not want the total discounted amount to exceed
N$3200.
Draw a graph and use it to answer the following questions:
i. How many tickets altogether should the organisers sell in order to maximise their sales
amount?
ii. How many tickets of each type (adult and child) should they sell to get a maximum
sales amount?
iii. What is the maximum sales amount the organisers can make?
iv. If the profit is calculated as follows: profit = total sales – total discount amount. How
much profit will the organisers make from the maximum sales?

A couple is planning a wedding and they need to decide how many guests on the bride’s side
as well as on the groom’s side to invite. The couple has agreed that the bride can invite at
most 60 guests. Since the bride’s friends and family are very fancy it costs N$300 per guest
on her side while it only costs N$150 per guest on the groom’s side and they have a budget
of N$30000. Each guest on the bride’s side will receive 4 drink tickets and each guest from
the groom’s side will receive 3 drink tickets and no more than 500 drink tickets can be given.
Keeping in mind that both the bride and groom must have guests (neither of them can attend
the wedding without any of their friends or family present).
Draw a graph and use it to answer the following questions:
i. What is the maximum number of guests that can attend the wedding?
ii. How many guests will the bride and the groom each invite?

Let f : R2-->R1be defined by
f(x1, x2) = (3x1+ 4x2)and T : R2--> R2be
defined by T(x1 , x2) = (x1- x2, x1+ x2).
Suppose g = f o T. What is g (2, 3) ?

Given the matrices (Each matrices are inside a [ ] )
A= 3 0 , B= 4 -1 , C= 1 4 2 , D= 1 5 2 , E= 0 1 3
-1 2 0 2 3 1 5 -1 0 1 -1 1 2
1 1 3 2 4 4 1 3

1.-4C^2
2.(E-D)^T
3. A(BC)
4. (B^T B)C
5. 3B-AB

Given the matrices (Each matrices are inside a [ ] )
3 0 1 5 2 0 1 3
A= -1 2 , B= 4 -1 , C= 1 4 2 , D= -1 0 1 , E= -1 1 2
1 1 0 2 3 1 5 3 2 4 4 1 3

Compute if possible
1. -4C^2
2. (E-D)^T
3. A(BC)
4. (B^T B)C
5. 3B-AB

A and X are the matrices
a b
c d
and
x y
u v
respectively, where b is not equal to zero. Prove that if AX = XA then u = cy/b and v = x + (d − a)y/b. Hence prove
that if AX = XA then there are numbers p and q such that X = pA + qI, and find
p and q in terms of a, b, x, y.

A and X are the matrices
a b
c d !
and
x y
u v !
respectively, where b is not equal
to zero. Prove that if AX = XA then u = cy/b and v = x + (d − a)y/b. Hence prove
that if AX = XA then there are numbers p and q such that X = pA + qI, and find
p and q in terms of a, b, x, y.

3. If A = i 0
0 i

where i =√−1, find A^2
and A^4.

Given that M =
2 −1
−3 4
and that M2 − 6M + kI = 0, find k
(Under matrix )

Consider A\\ =\\left[\\begin{array}{cc} 2 & 3\\\\ 1 & 2\\end{array}\\right],\\ B\\ =\\left[\\begin{array}{cc} 1 & 5\\\\ \\end{array}\\right]; what is the identity element of the two matrices
a.\\(\\left[\\begin{array}{cc} 0 & 1\\\\ 1 & 0 \\end{array}\\right]\\)
b.\\(\\left[\\begin{array}{cc} 1 & 0\\\\ 0 & 1 \\end{array}\\right]\\)
c.\\(\\left[\\begin{array}{cc} 0 & 0\\\\ 1 & 1 \\end{array}\\right]\\)
d.\\(\\left[\\begin{array}{cc} 2 & -2\\\\ -13 \\end{array}\\right]\\)