Lecture 11' Matrix inversion and linear equations

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Lecture 11. Matrix inversion and linear equations

• Learning objectives. By the end of this lecture
  you should
• Know more about determinants and matrix inversion
• Understand how linear equations can be
  represented in matrix form
• Know how to solve linear equations using
  matrices.
• Introduction Inverses and determinants -
  reminder.
• The concept of inverse only applies to nxn
  matrices.
• If it exists, the inverse of A is a unique matrix
  A-1 such that AA-1 I A-1A
• The inverse does not exists if and only if the
  determinant of A 0.
• To calculate A-1
• First check that det. A ? 0
• Then calculate the matrix of minors, M
• Then from this calculate the matrix of co-factors
  C
• Then A-1 (1/det. A)CT.

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2. Example

• Find the matrix of minors.
• Find the determinant
• Find the inverse and check it.

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3. Linear Equations

• Recall that linear equations are those where the
  variables enter in a linear form
• e.g.
• 4 x1 x2 is linear
• 4 x2x1 x2 is not linear because of the x2x1
  term.
• 4 x1 x2 can be written in terms of vector
  products

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4. Matrix form for linear equations

• If there are several simultaneous linear
  equations, they can be stacked
• e.g.
• 4 x1 x2
• 3 2x1 x2
• In general, a system of m linear equations in n
  unknowns can be represented as
• Ax b
• Where A is an mxn matrix, x is a nx1 matrix (or
  column vector) and b is an mx1 matrix.
• Conversely Ax b can be interpreted as a system
  of m linear equations in n unknowns.

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5. Solving linear equation systems

• In general we want to find the solution to the
  equation Axb.
• There are three possible cases
• There is no solution.
• There is exactly one solution
• There is more than one solution (typically an
  infinite number).
• It is not always obvious which case applies.
• Example 1. No solution.
• This says that 4 x1 x2 and 3 x1 x2
  simultaneously. Thus 3 4, which is nonsense.
  Hence there is no solution.

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5. Solving linear equation systems

• Example 2. One solution.
• This says that
• 4 x1 x2 and
• 3 2x1 x2.
• Thus -1 x1 so x2 5.
• Example 3. Many solutions.
• This says that
• 4 x1 x2 and
• 8 2x1 2x2.
• So, all we can say is that x2 4-x1. If x1 0,
  then x2 4 if x1 1 then x2 3 etc.

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6. Solving when there are n equations and n
unknowns

• This is a special case. In general we may have
  more equations than unknowns or vice versa.
• If n unknowns then m n so in the equation Ax
  b, A is a square matrix.
• One solution. Suppose det. A ? 0 so that A-1
  exists.
• If Ax b then A-1Ax A-1b
• Or x A-1b and we have our solution.
• Example
• So,

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6. Solving when there are n equations and n
unknowns

• No solution. Suppose b ? 0 then if there is no
  solution A-1 does not exist i.e. det. A 0.
• Example
• So det. A 1 -1 0.

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6. Solving when there are n equations and n
unknowns

• Many solutions. Suppose b ? 0 then if there are
  multiple solutions A-1 does not exist i.e. det.
  A 0.
• Example
• So any x2 4-x1 is a solution. In this case.
  Det. A 2-20.
• Compare 2) and 3). When there are no solutions
  det. A 0 and when there are multiple solutions,
  det. A 0. So when det. A 0 all we know is
  that there isnt one solution. We dont know if
  were in the no solution or the multiple
  solutions case.
• To find out, we need to check to see if the
  equations are consistent. In case 2. the
  equations are inconsistent hence there are no
  solutions. In case 3, the equations are
  consistent hence there are many solutions.

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Quiz II.

• Use matrix inversion to find the solution to the
  following set of equations.
• Why in this case is it quicker not to use matrix
  inversion?

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7. Practical tips on finding determinants

• If the matrix is square then when seeking a
  solution for x the first issue is whether the
  determinant is zero.
• In short, if the matrix is square you need first
  to find its determinant.
• Example
• So, A-1 exists. In fact,
• Thus

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7. Practical tips on finding solutions

• Whats the solution to this set of linear
  equations?
• Obviously this is related to the last set of
  equations - x1 has been relabelled as x3 and vice
  versa. So,
• Notice that while A-1 simply involves relabelling
  rows of the previous inverse, det. A is actually
  different
• Det. A 2(0-0)-1(-1-0) 0(-1-0)1

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7. Practical tips on finding solutions

• General properties of determinants.
• If B AT, then det. B det. A.
• If B is constructed from A by swapping two rows,
  then det. B -det. A
• If B is constructed from A by swapping two
  columns, then det. B -det. A
• If B is constructed from A by multiplying one row
  (or column) by a constant, c, then det. B c
  det. A
• If B is constructed from A by adding a multiple
  of one row to another, then det. B det. B.

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7. Practical tips on finding solutions

• Some implications of properties of determinants.
• You can use any row to calculate the determinant
  (often there are easy rows to use).
• Using row 1, det. A 12(2-0)4(00)6(0-3) 6
• Using row 2, det. A 1 (12.2-3.6) 6
• So
• If one row (column) is a multiple of another row
  (column) then det. A 0.
• If one row (column) can be constructed by
  adding/subtracting multiples of other rows
  (columns) then det. A 0.

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Quiz III.

• Show that all the determinants equal 0

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8. Summary.

• 3 skills you should be able to do
• Write down a system of linear equations in matrix
  form.
• Check whether a system of n equations in n
  unknowns has one solution.
• Find short cuts for the determinant of a square
  matrix