cramer’s rule

Cramer’s RuleCramer’s RuleVIVIANA MARCELA BAYONA CARDENAS

Coefficient MatricesCoefficient Matrices

You can use determinants to solve a system of linear equations.

You use the coefficient matrix of the linear system.

Linear System Coeff Matrixax+by=e

cx+dy=f

dc

ba

Cramer’s Rule for 2x2 Cramer’s Rule for 2x2 SystemSystem

Let A be the coefficient matrix Linear System Coeff Matrix

ax+by=ecx+dy=f

If detA 0, then the system has exactly one solution:

A

df

be

xdet

and

A

fc

ea

ydet

dc

ba

Example 1- Cramer’s Rule Example 1- Cramer’s Rule 2x22x2

Solve the system:8x+5y=22x-4y=-10

42

5842)10()32(

42

58

The coefficient matrix is:and

So:

42

410

52

xand

42

102

28

y

142

42

42

)50(8

42

410

52

x

242

84

42

480

42

102

28

y

Solution: (-1,2)

Learning objectives. By the end of this lecture you should:

◦ Know Cramer’s rule◦ Know more about how to solve linear equations

using matrices.

1. Introduction: Cramer’s rule. Often when faced with Ax=b we are not

interested in a complete solution for x.We may only wish to find x1 or x4Cramer’s rule is a short cut for finding a

particular xi. It’s particularly useful when A is 3x3 or bigger.

It is not sensible to use it if you need to find several xis – finding A-1 is then generally quicker.

Suppose you have the system of equations,Ax = b.

Define the matrix Ai as the result of replacing in the ith column of A with b:

Example 1.

Example 2.

Cramers ruleCramers rule

nnnnn

n

b

b

x

x

aa

aa

11

1

111

nn

n

ba

ba

A

1

111

13

14

3

4

12

111

2

1 Asox

x

Suppose you have the system of equations,Ax = b, then, if det. A≠ 0,

Example 2. (Recall that the solution to this system was x1 = -1, x2 = 5.)

So x1 = 1/-1 = -1 and x2 = -5/-1 = 5.

2. Cramers rule2. Cramers rule

A

Ax ii

532

41det;1

13

14det1

12

11.det 21 AAA

Cramer’s rule in macroeconomicsCramer’s rule in macroeconomics

Many macro models involve a system of linear equations.

Cramer’s rule can be used to solve for one particular variable.

Example. Suppose 1. Y = C + I + G2. C = a + bY 0 < b < 13. I = I04. G = G0Write this system in matrix form then use Cramer’s rule to

find consumption, C.

Step 1: identify the endogenous variables and the exogenous variables. The endogenous variables correspond to the x vectors in the previous example. The exogenous variables (the parameters of the system) correspond to the b vector.

Example: here C and Y are endogenous. I0 and G0 are the exogenous variables.


Step 2: Simplify the system of equations if possible then write down the system in such a way that all the endogenous variables are on one side of the equation and all the exogenous variables are on the other side.

Example: simplify the equations1.Y = C + I0 + G02.C = a + bY.Rewrite:1.Y - C = I0 + G02.C – bY= a

Step 3. Put into matrix form.Matrix form:

a

GI

C

Y

b00

1

11


Step 4. then use Cramer’s rule

So, to find C we replace the second column of the matrix with the column vector of parameters.

Quiz II. Find Y using the same procedure.

a

GI

C

Y

b00

1

11

b

GIba

b

ab

GI

C

1

)(

1

11

1

00

00

Some guidance on solving mxn equation Some guidance on solving mxn equation systems.systems.The general problem involves m equations and n

unknowns.Many systems of equations involve fewer

equations than variables, m<nSome involve more equations than variables, n <

m. In either case you cannot use matrix inversion to

characterise the solution (if it exists).

Example.

When m ≠ n we seek to do two things:1. Find out if any solution exists.2. If at least one solution exists, identify its features.

3

2

1

101

111

1

0

x

xx

Some guidance on solving mxn equation Some guidance on solving mxn equation systems.systems.Definition: The rank of a matrix is the largest

number of linearly independent rows or columns.

Note that the column rank and the row rank will be the same.

Note that the rank cannot be larger than the smaller of m and n. i.e. if A is an mxn matrix rank(A) ≤ min(m,n)

Example.

The rank of this matrix is at most 2, but in fact rank(A) = 1.

The rank of a matrix provides a guide to number of solutions.

111

111A

Some guidance on solving mxn equation Some guidance on solving mxn equation systems.systems.Note that for an nxn matrix (det A = 0) ↔ rank(A) < n.

We can see ← from the properties of determinants. If rank(A) < n we can add and subtract rows to create a row of zeros. The determinant of this new matrix is therefore 0, but by property 5 adding and subtracting rows does not change the determinant. So det(A) = 0.

Example. A obviously has rank of less than 3 because the third row equals the sum of the two other rows. What is its determinant?

110

120

010

A

Some guidance on solving mxn equation Some guidance on solving mxn equation systems.systems.

1. Find the rank of the system. Note that the maximum possible rank is n.i. If rank(A) = n, then there may be a unique

solutionii. If rank(A) < n then there cannot be a

unique solution.

2. Check consistency (i.e. the absence of contradictions)i. If the system is consistent and rank(A) = n

then there is exactly one solution. ii. If the system is consistent and rank(A) < n

then there are multiple solutions.

REFERENCEREFERENCE

cramer’s rule

Documents