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Matrices Chapter 5


A calculator could do ones with numbers in! Simultaneous equations. Equations like ... If using 'algebra' eliminate the same unknown between 2 pairs of equations ... – PowerPoint PPT presentation

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Title: Matrices Chapter 5

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

Matrices (Chapter 5)
  • A singular matrix has determinant 0
  • If M is non-singular it has an inverse M-1 given
  • (MN)-1 N-1M-1
  • Using matrices to solve simultaneous equations,
    and the geometric interpretation

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

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

Inverse of a 3 x 3 matrix
  • Example
  • det M
  • Cofactors are

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!

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

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

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.

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

Try it with the matrix on the previous page!
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)
Questions Winter 06
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
  • (vi) Quite well known. Dont forget I

Questions Summer 06
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)

There will now be a short break
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