1 / 18

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

by

- (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

so

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

and

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

D - (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