Matrices Chapter 5

1
Matrices (Chapter 5)

• Ideas from Further Pure 1
• The 2 x 2 matrix M is

• Representing a 2d transformation by a 2 x 2
  matrix (columns are images of (1,0) and (0,1))
• Multiplying matrices (rows by columns)
• Identity matrices (2 x 2 and 3 x 3)
• The determinant of the matrix M is ad bc
• det M is the signed area scale factor of the
  transformation represented by M

2
Matrices (Chapter 5)

• A singular matrix has determinant 0
• If M is non-singular it has an inverse M-1 given
  by

• (MN)-1 N-1M-1
• Using matrices to solve simultaneous equations,
  and the geometric interpretation

3
Inverse of a 3 x 3 matrix
sign minor cofactor

• (expansion by the first column)
• Some like Sarrus method (see textbook)
• If two rows or columns are the same, the
  determinant is zero

4
Inverse of a 3 x 3 matrix

• det(MN) det M x det N
• To find the inverse
• Find det M
• Find the adjugate matrix (the transpose of the
  matrix of cofactors)
• Divide this by the determinant

5
Inverse of a 3 x 3 matrix

• Example

• det M
• Cofactors are

6
Inverse of a 3 x 3 matrix

• Adjugate matrix is
• Inverse is

• The inverse exists unless k 9.5
• A calculator could do ones with numbers in!

7
Simultaneous equations

• Equations like
• represent planes

• How to solve them?
• Try a matrix, but it may be singular
• By algebra eliminate the same unknown between
  two pairs of equations

8
Simultaneous equations

• Geometrical interpretation
• If the matrix has an inverse, the three planes
  meet in a unique point
• If the matrix is singular, the equations could be
  inconsistent (no solution)
• triangular prism (no two parallel)
• various possibilities if planes are parallel
• Or consistent (infinitely many solutions)
• line of common points (sheaf)
• all three planes are coincident

9
Eigenvalues and eigenvectors

• If s is a non-zero vector so that Ms ?s
• s is an eigenvector of M with eigenvalue ?
• Points on lines defined by eigenvectors stay on
  those lines
• Finding eigenvectors
• Ms ?s ? (M ?I)s 0
• which must have a non-zero solution for s
• so det(M ?I) 0 the characteristic eqt.

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Eigenvalues and eigenvectors
Char. eqt.
so the eigenvalues are -3 and -4. To find the
eigenvector for -3,
or any multiple is an eigenvector.
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Eigenvalues and eigenvectors
Algebra gives eigenvalues 1, 2, 3. Evector for 1
so
is an eigenvector the others are
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Matrix algebra

• Form the diagonal matrix ? of evalues
• and the corresponding matrix S of evectors
• Then M S?S-1
• Use Finding powers of matrices

and
Try it with the matrix on the previous page!
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Matrix algebra

• The Cayley-Hamilton Theorem
• A matrix satisfies its own characteristic eqt.

(trust me)
Char. eqt.
(must have I)
C-H ?
We can use this to find M-1
(must have I)
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Questions Winter 06
15
Examiners Report

• A very good source of marks
• (i) Very well done
• (ii) Little trouble in finding eigenvalues
• (iii) Little trouble in finding eigenvector
• (iv) Only have to verify
• (v) Some did not know this. Others forgot to cube
  D
• (vi) Quite well known. Dont forget I

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Questions Summer 06
17
Examiners Report

• A completely different matrix question
• (i) Amazing how good candidates are at this.
  Watch the arithmetic, though
• (ii) Can use (i) but many missed the link
• If using algebra eliminate the same unknown
  between 2 pairs of equations
• (iii) Found very challenging the best way is as
  above (same unknown, 2 pairs)

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There will now be a short break