For the matrix A = 2 41 2 1 0 0 3 1 0 13 5

For the matrix A = 2 41 2 1 0 0 3 1 0 13 5

Subject: Mathematics    / General Mathematics
Question

Examples:

A. For the matrix A = 2 41 2 1 0 0 3 1 0 13 5

(i) calculate det(A) and deduce that A is invertible;

(ii) use row operations to calculate A-1;

(iii) evaluate det(A-1) and verify that det(A-1) × det(A) = 1.

B. (Cost allocation model) The company accountant allocates costs by separating costs into direct costs and internal costs for each department. The

development department charges 10% of its total monthly costs (which are

$x) to the promotional department. The promotional department charges

5% of its total monthly costs (which are $y) to the development department. The direct costs for the development department are $20,400 and for

the promotional department they are $9,900. Hence the following matrix

equation must hold:

20400 9900 + 00 0 :1 0 :05 xy = xy :

Find the total costs for each department.

Questions:

1: Let A = 2 4a b c d e f g h i3 5 : Assuming that det(A) = 8, find:

(a) det(2A) (b) det(A-1) (c) det(2A-1) (d) det[(2A)-1]

(e) det

24

a d g

b e h

c f i

35

(f) det

24

a b c

g h i

d e f

35

(g) det

24

d e f

a=2 b=2 c=2

g h i

35

2: For the matrix A = 2 44 0 2 1 2 1 0 3 13 5

(a) calculate det(A) and deduce that A is invertible;

(b) use row operations to calculate A-1;

(c) evaluate det(A-1) and verify that det(A-1) × det(A) = 1;

(d) use the matrix inverse A-1 to solve the system of linear equations:

4x + 2z = 6 ;

x + 2y + z = 1 ;

3y + z = 3 :

3: (a) Under what conditions is the matrix C =

24

a 0 0

0 b 0

0 0 c

35

invertible?

Find C-1 and det(C-1).

(b) When are the triangular” matrices A =

24

a 0 0

d b 0

e f c

35

and B = 2 4a g h 0 0 0b i c3 5

invertible?

4: A town has four industries: agriculture, fishing, lumber and textiles. For a

Leontief open economic model, the input-output matrix for this economy is

A =

2664

0:3 0:1 0:1 0:6

0:1 0:3 0 0:1

0:3 0:4 0:2 0:2

0:1 0:2 0:1 0:1

3775

where the columns represent the inputs of agriculture, fishing, lumber and

textiles, respectively.

(a) Which industries are profitable?

(b) Which industries break even?

(c) What inputs does the fishing industry require from all four industries

to make $1000 of fishing products?

(d) What is the difference between the fishing industry’s output and inputs?

(e) How much profit does the fishing industry make for each dollar of output?

(f) What inputs does the lumber industry require from all four industries

to make $1000 of lumber products?

(g) What is the difference between the lumber industry’s output and inputs?

(h) How much profit does the lumber industry make for each dollar of output?

5: The engineering department of a firm charges 20% of its total monthly costs

to the computer department and the computer department charges 30% of

its total monthly costs to the engineering department. Say the total monthly

costs of the engineering department is $x and the total monthly costs of the

computer department is $y. Say, during a given month, the engineering

department’s costs, not including what it paid to the computer department,

were $94; 000, and the computer department’s costs, not including what it

paid to the engineering department, were $112; 000.

(a) Write equations for x and y.

(b) For x = (x; y) and d = (94000 ; 112000), determine the matrix A in the

equation x = d + Ax.

(c) Calculate the total monthly costs for each department, to the nearest

dollar.

6?: For the matrix A = 2 4 6 -3 find A-1 as products of elementary matrices