ECON 2201 University of Winnipeg Matrix Questions

Description

Try to explain first in English what you are doing, before jumping into formulas. For example “Next, I
will calculate the cofactors of each element of the coefficient matrix”, “I will expand the determinant
by the third row”, etc….
You are answers should be in ONE PDF file (that is ONE file, and it is not a JPG, HEIC or whatever
other format).

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Due date: Whatever Nexus says.
Try to explain first in English what you are doing, before jumping into formulas. For example “Next, I
will calculate the cofactors of each element of the coefficient matrix”, “I will expand the determinant
by the third row”, etc….
You are answers should be in ONE PDF file (that is ONE file, and it is not a JPG, HEIC or whatever
other format). All pages in your file have to be in portrait orientation. Pages that are rotated 90 (or any
other angle except zero) degrees will not be be marked.
No submissions will be accepted after the answer key becomes available!
Hint: for question 1 and 2 you might want to read (carefully!) pages 283-286 of the textbook.
Consider an economy having three industrial sectors: light industry (its output denoted by y1),heavy
industry(its output denoted by y2) and agriculture (its output denoted by y3). There are three factors of
production: labour (l), capital (k), and energy (e). The input requirement matrix is given by
3 1 2
A= 2 5 1
1 1 3
This shows that, for example, to produce one unit of light industry the country need to use 3 units of
labour, 1 unit of capital, 2 units of energy. Similarly, one unit of heavy industry is produced by using 2
units of labour, 5 of capital and 1 of energy, and so on.
1. How many units of inputs are required to produce an output of one unit of light industry, two
units of heavy industry and three units of agricultural goods? Use matrix algebra to answer this
question. Hint: the answer is a multiplication of a matrix and a column vector.
2. The country has 100 units of each input. If it fully uses all the inputs, what would be the output
of the country? Use matrix algebra and an inverse matrix to answer this question.
[ ]
An economy is described by the following equations:
Y =C+ I 0 +G 0+ NX
C =a+ b(Y −T )
T =d +tY
NX = fY −g E
where a, b, d, t, f and g are positive parameters. Y is the GDP, C is private consumption, G 0 is
government’s consumption, NX is net exports, T is taxes and E is the real exchange rate.
Assume that this country uses a fixed exchange rate regime (thus E is exogenously given).
3. Write the above system of equation in matrix form. Consider the vector of endogenous variable
to be X= [ Y, C, T, NX]T.
4. Under which condition(s) does this system have a unique solution?
5. Solve the system by using matrix inversion.
6. Find the equilibrium value of GDP by using Cramer`s rule. Check that you get the same answer
as for question 5.
7. How would you change your answer to question 3 if you want to find the value of E that keeps
the GDP equal to its potential level, Y? Hint: you will have to solve the system for E, and that
means the vector of endogenous variables will change a bit from the one from question 3.
Consider the quadratic form
Q = -2×12 + 4x1x2 – 5×22 + 2x2x3 – 3×32 + 2x1x3
8. Write down the quadratic form withe the help of matrix, in the form XTAX, where
X = [x1 x2 x3]T
9. Determine the “definitness” of matrix A.
[
]
2 2
.
2 −1
10. Find the characteristic roots of this matrix.
11. Using the characteristic roots, determine the “definitness” of this matrix.
Consider the matrix

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Tags:
matrix algebra

units of energy

Heavy industry

agricultural goods

Light industry

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