Linear Algebra I

1.(10 points) Given the following system of linear equations

4

3

14

16

4

2

7

8

2

3

2

3

2

1

3

2

1

=

+

=

+

+

=

+

+

x

x

x

x

x

x

x

x

(a) Solve the following system of equations by performing elementary row operations on

the augmented matrix.

(b) Based on your answer for (a) what can you say about the invertibility of the

coefficient matrix? Justify your answer.

2. (10 points) Consider the linear system (where  and  are constants):

3

4

3

2

1

3

=

+

+

−

=

+

+

=

+

+

z

y

x

z

y

x

z

y

x





Find conditions on  and/or  such that:

(a) The system has a unique solution.

(b) The system has no solution.

(c) The system has infinitely many solutions.

3.(30 points) Construct a consumption matrix for an open productive economy consisting of

three sectors. Use the Leontief Input-Output Model to show that the economy is productive. Do

Provide the written explanation for question 3 for presentation