Lecture 12' Solving linear equations

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Lecture 12. Solving linear equations

• Learning objectives. By the end of this lecture
  you should
• Know Cramers rule
• Know more about how to solve linear equations
  using matrices.
• Introduction Cramers rule.
• Often when faced with Axb we are not interested
  in a complete solution for x.
• We may only wish to find x1 or x4
• Cramers rule is a short cut for finding a
  particular xi. Its 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.

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2. Cramers rule

• 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.

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2. Cramers rule

• 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.

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3. Quiz 1.

• Suppose,
• Use Cramers rule to find x1 and x2 .

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4. Cramers rule in macroeconomics

• Many macro models involve a system of linear
  equations.
• Cramers rule can be used to solve for one
  particular variable.
• Example. Suppose
• Y C I G
• C a bY 0 lt b lt 1
• I I0
• G G0
• Write this system in matrix form then use
  Cramers 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.

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4. Cramers rule in macroeconomics

• 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 equations
• Y C I0 G0
• C a bY.
• Rewrite
• Y - C I0 G0
• C bY a
• Step 3. Put into matrix form.
• Matrix form

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4. Cramers rule in macroeconomics

• Step 4. then use Cramers 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.

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6. Some guidance on solving mxn equation systems.

• The general problem involves m equations and n
  unknowns.
• Many systems of equations involve fewer equations
  than variables, mltn
• Some involve more equations than variables, n lt
  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
• Find out if any solution exists.
• If at least one solution exists, identify its
  features.

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6. Some guidance on solving mxn equation 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.

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6. Some guidance on solving mxn equation systems.

• Note that for an nxn matrix (det A 0) ? rank(A)
  lt n.
• We can see ? from the properties of
  determinants. If rank(A) lt 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?

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6. Some guidance on solving mxn equation systems.

• Find the rank of the system. Note that the
  maximum possible rank is n.
• If rank(A) n, then there may be a unique
  solution
• If rank(A) lt n then there cannot be a unique
  solution.
• Check consistency (i.e. the absence of
  contradictions)
• If the system is consistent and rank(A) n then
  there is exactly one solution.
• If the system is consistent and rank(A) lt n then
  there are multiple solutions.

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6. Some guidance on solving mxn equation systems.

• Example
• This has a rank of 2. We can take the second row
  away from the first row to create
• Or x1 1-2x3 and x2 13x3. This is as far as
  we can go in defining a solution. One variable is
  free to take on any value which then determines
  the value of the other two variables. For
  example, given any x3 we can calculate the other
  two variables.
• In general, if the equation system is consistent,
  then the number of free variables is n-rank(A).

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6. Some guidance on solving mxn equation systems.

• Example 2
• Here m 3 and n 2, so we might suspect that
  the system is contradictory.
• From the third equation we get x2 2
• But that tells us that (from the first equation)
  x1 -2.
• Now insert these results into the middle equation
  and we get
• 2x1 - x2 -6 1. This is obviously wrong, so
  there is no solution.
• Notice that if the equation system was
• Then there would be a unique solution.

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Summary.

• 3 skills you should be able to do
• Use Cramers rule to solve for a variable
• Write down a linear macroeconomic system in
  matrix form
• Work out the rank of a matrix
• Characterise the solution of Axb.