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Ch. 4 Linear Models & Matrix Algebra

• Matrix algebra can be used:• a.       To express the system of

equations in a compact manner.• b.      To find out whether solution

to a system of equations exist.• c.       To obtain the solution if it

exists.

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4.1 Matrices and VectorsMatrices as ArraysVectors as Special Matrices

• Matrix is a rectangle array of parameter, coefficients, etc.

• A general form matrix Ax = d,

mn

n

n

mn

n

n

mm d

d

d

x

x

x

ax

ax

ax

axa

axa

axa

2

1

2

22

2

211

22121

2111

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Step 1: Write in matrix format:

mn

n

n

mn

n

n

mm d

d

d

x

x

x

a

a

a

aa

aa

aa

2

1

2

21

2221

211

A x = d

A = parameter matrixx = variable column vectord = constant column vectorA general form matrix Ax = d, solve for x;

4

1

21

22221

11211

nnnn

n

n

aaa

aaa

aaa

nx

x

x

2

1

nd

d

d

2

1

Solving for X

x = A-1 d ,

where A-1 is the inverse (matrix) of A

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Inverse A-1of Matrix of A

• Inverse of A is A-1 • AA-1 = A-1A = I• We are interested in

A-1 because x=A-1d

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Derivation of matrix inverse formula

A-1 = adjoint A / |A|, where

|A| = ai1ci1 + …. + aincin (Determinant)

And, adjoint A = transposed cofactor matrix of A

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Determinant, Cofactor, and Minor

3231

222113

3331

232112

3332

232211

aa

aaM

aa

aaM

aa

aaM

333231

232221

131211

aaa

aaa

aaa

A

ijji

ij MC 1

n

jjj CaA

111

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How to get Determinant of A?

jj

j

n

j

n

jjj MaCaA

11

111

11 )1(

By Laplace Expansion of cofactors, and minors in case the first row is used.

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• Pattern of the signs for cofactor minors

:1__ jionfocus

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Adjoint of A: the transposed cofactor matrix

mmm

n

n

nxn

aaa

aaa

aaa

A

21

22221

11211

mnmmi

n

x

CCC

CCC

CCC

C

2

22221

11211

Aadj

CCC

CCC

CCC

C

nnnn

m

m

nxn

21

22212

12111

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Calculating Adjoint is hard!

Is there any easier way to solve for x or specifically one of x, that is, xi ?

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Cramer's Rule for each of x, say, x1 : “The easy way”

A

Ax i

i

•The numerator represents a determinant of A in which the ith column is replaced by the vector of constants, i.e., no need to invert A!

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Solving for x1 using Cramer’s rule

• Find the determinant |A|• Find the determinant |A1|

where di is the constant vector substituted for the 1st col.• X1 = |A1|/|A|• Repeat for X2 by substituting the constant vector for the 2nd

col. And solving for |A2| and so on as necessary

mmm

n

n

nxn

aaa

aaa

aaa

A

21

22221

11211

nnnn

n

n

nxn

aad

aad

aad

A

2

2222

1121

1

14

1

21

22221

11211

nnnn

n

n

aaa

aaa

aaa

nn dx

dx

dx

/

/

/

22

11

1

12

/

/

1

dd

dd

n

Solving for x1 / d1

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What about Comparative Statics?

333231

232221

131211

aaa

aaa

aaa

A