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each column can contain at most one pivot, there is a pivot in each column. So,

every variable is a basic variable; there are no free variables. The reduced row

echelon matrix AвЂќ has the form

If there is a solution for some given RHS h,, , b,,,, it will be unambiguously

determined by AвЂќ; that is, the solution will bc unique.

(Only If): On the other hand, if the rank is less than the number of columns,

then there must be some free variables. Choose a RHS so that the system has a

solution, for example, h, = = b,,, = 0. Because the fret variables can take

on any value in solutions (as shown in the previous section), there arc infinitely

many solutions to the system. This proves the second half of Fact 7.9. n

Finally, we combine Facts 7.7 and 7.9 to characterize those coefficient matrices

which have the property that for any RHS hi,. , b,,,l the corresponding system

of linear equations has exactly one solution. Such coefficient matrices are called

nonsingular. They are the ones which will arise most frequently in our study ot

linear systems and other linear phenomena.

Fact 7.10. A coefficient matrix A is nonsingular, that is, the corresponding

linear system has OK and only one solution for every choice of right-hand side

b,, , b ,,,, if and only if

number of rows of A = numhcr of columns ofA = rank,4

Fact 7.10 is a straightforward consequence of Facts 7.7 and 7.9. It tells us that

a necessary condition for a system tn have a unique solution for every RHS is that

there be exactly as many equations as unknowns. The corresponding coefficient

matrix must have the same number of rows as columns. Such a matrix is called a

square matrix.

a square maximal rank

The problem of determining whether matrix has (that

is, rank as in Fact 7.10) is a central one in linear algebra. Fortunately, there is

an easily computed number which one can assign to any square matrix which

determines whether or not this rank condition holds. This number is called the

determinant of the matrix; it will be the subject of our discussion in Chapters 9

and 26.

Finally, Fact 7.1 I summarizes our findings in this section for a system of m

equations in II unknowns-a system whose coefficient matrix has m rows and n

columns.

b

Fact 7.11. Consider the linear system of equationsAx =

(u) If the number of equations < the number of unknowns, then:

(i) Ax = 0 has infinitely many solutions,

b; Ax b

(ii) f<)r any given = has 0 or infinitely many solutions, and

(iii) if rank A = number of equations, Ax b

= has infmitcly many solu-

tions for every RIIS b.

(/I) If the numhcr of equations > the number of unknowns, then:

(i) Ax = 0 has one or infinitely many solutions,

(ii) for any given b, Ax = b has I), l_ or infinitely many solutions, and

Ax

(iii) if rank A = number of unknowns, = b has 0 or I solution for

cwry RHS b.

(c) If the number of equations = tbc number of unknowns, then:

(i) 4x = 0 has one or infinitely many solutions.

b, b

(ii) for any given Ax = has 0, I, or infinitely many solutions, and

A Ax = b

(iii) if rank = number of unknowns = number of equations.

has rxi˜tly I solution for every RHS h.

Application to Portfolio Theory

duplicable

A portfulic (1,. .,I˜,) is called if there is a different portfolio

(M.,, br:,) with exactly the came returns in every state:

148 SYSTEMZ OF LINEAR EqLlATloNs [71

A state s* is called insurable if there is a portfolio (.x1, , .q) which has a positive

return if state se вЂњccurs and zerвЂќ return if any other state вЂњCCUTS:

For any portfolio x, the return to x in each state is given by the S-tuple (RI, , Rs),

where

: : (22)

Rsl xl + + Rsa xa = Rs.

Let R be the S X A coefficient matrix of the RsiвЂ™s:

Suppose first that the matrix R has ranks = the number of rows of R. Then, by

Fact 7.7, one can solve system (22) for any given S-tuple (RI,. , Rs) of returns.

IвЂќ particular, if we take RI = = Rs = b for sвЂќme b # 0, the solution to (22),

when properly normalized so that x, + + X˜ = 1, will be a riskless asset. If we

set Rx = I and K, = 0 for i # k, the solution tвЂќ (22), when properly normalized,

will be aвЂќ insurance portfolio for state k. So, if the rank of R = S, then there is a

riskless asset and every state is insurable.

We will argue in Section 2X.2 that the converse holds toвЂќ. If every state is

insurable, then R must have rank S. In particular, if A < S, that is, if there are

more states of nature than assets, then I( cannвЂќt have rank S and there must exist

states that are not insurable.

Finally. there are duplicable portfolios if and only if equation (22) has multiple

portfolio solutions for some right-hand sides. This situation occurs only if system

(22) has free variables, that is, only if the rank of R is less thanA.

Example 7.6 In Example 6. I, we worked with the 3 X 2 state--return matrix

()

13

R= 2 2,

3I

149

L7.41 RANK-THE FUNDAMENTAL CRITERION

which has 2 columns and rank 2. We use Gaussian elimination to transform ?R

I .вЂ˜:

to ˜5 tow echelon form:

26 = 0 if a = b = c = 1, this market has a riskless asset.

Since a + c

Since a + c ˜ 26 # 0 if (a, b, c) has exactly one nonze˜o component, there

are no insurable states. Since X has no free variables, there are no duplicable

portfolios.

EXERCISES

7.20 Compute the rank of each of the following matrices:

6 -7 1

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