Further Pure 4 Notes : Robbie Peck

Matrix Algebra

𝐴 𝐵 + 𝐶 + 𝐴𝐵 + 𝐴𝐶, 𝐵 + 𝐶 𝐴 = 𝐵𝐴 + 𝐶𝐴, 𝐴 𝐵𝐶 = 𝐴𝐵 𝐶

(𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇 where T denotes a transposed matrix

(𝐴𝐵)−1 = 𝐵−1𝐴−1 where A-1 denotes the inverse matrix of A.

Transformations

𝑎 𝑏𝑐 𝑑

⇒ The image of 𝑖 is 𝑎𝑐

and image of 𝑗 is 𝑏𝑑

𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 𝑕 𝑖

⇒ The image of 𝑖 is

𝑎𝑑𝑔

and image of 𝑗 is 𝑏𝑒𝑕

and of 𝑘 𝑐𝑓𝑖

There are many standard transformation matrices, most of which are in the formula book.

0 1 01 0 00 0 1

, 1 0 00 0 10 1 0

and 0 1 01 0 00 0 1

are reflections in 𝑥 = 𝑦, 𝑦 = 𝑧 and 𝑥 = 𝑧

Vectors

𝑎 ∙ 𝑏 = 𝑎 𝑏 cos 𝜃 is used to find the angle between two vectors 𝑎 and 𝑏.

𝑎 × 𝑏 = 𝑎 𝑏 sin 𝜃 𝑛 is used to find 𝑛 , the vector perpendicular to 𝑎 and 𝑏.

⇒ If 𝑎 × 𝑏 = 0, then ⇒ sin 𝜃 = 0 and ⇒ 𝜃 = 0 + 180𝑛 ⇒ 𝑎 and 𝑏 are parallel.

For vectors 𝑎 and 𝑏 𝑏 × 𝑎 = −𝑎 × 𝑏

Area of a triangle 1

2 𝑎 × 𝑏 for two vectors a and b

Area of a Parallelogram 𝑎 × 𝑏 for two vectors a and b

𝑎 ∙ 𝑏 × 𝑐 = 𝑎 × 𝑏 ⋅ 𝑐 = 𝑎 × 𝑐 ∙ 𝑏

Parallelepiped volume 𝑎 ∙ 𝑏 × 𝑐

Determinants

𝑀 is negative if the transformation 𝑀 involves a reflection.

𝑎 × 𝑏 = 𝑑𝑒𝑡 𝑖 𝑗 𝑘𝑎1 𝑎2 𝑎3

𝑏1 𝑏2 𝑏3

and 𝑎 ∙ 𝑏 × 𝑐 = 𝑑𝑒𝑡

𝑎1 𝑎2 𝑎3

𝑏1 𝑏2 𝑏3

𝑐1 𝑐2 𝑐3

Rules of determinant manipulation

1. 𝑀 = 𝑀𝑇

2. adding any multiple of a row (or column) to another row (or column) does not alter 𝑀 .

3. Interchanging two rows (or columns) does not alter 𝑀 .

4. Multiplying a row (or column) by a scalar multiplies 𝑀 by that scalar

𝐴𝐵 = 𝐴 𝐵

Application of vectors and determinants

Volume of a cuboid 𝑎 ∙ 𝑏 × 𝑐

Volume of a parallelepiped 𝑎 ∙ 𝑏 × 𝑐

Volume of a pyramid 1

3 𝑎 ∙ 𝑏 × 𝑐

Volume of a tetrahedron 1

6 𝑎 ∙ 𝑏 × 𝑐

Volume of triangular prism 1

2 𝑎 ∙ 𝑏 × 𝑐

If 𝑎 ∙ 𝑏 × 𝑐 = 0 then the vectors are on the same plane; a, b & c are coplanar.

Lines and Planes

Lines: Parametric format: 𝑟 = 𝑎 + 𝜆𝑏 Vector Product: 𝑟 − 𝑎 × 𝑏 = 0

Direction ratio: 𝑥−𝑎1

𝑏1=

𝑦−𝑎2

𝑏2=

𝑧−𝑎3

𝑏3 (Can be derived from vector product)

𝑏𝑖

𝑏 =

𝑏𝑖

𝑏12+𝑏2

2+𝑏32 is the direction cosine of the line. The sum of the squares is 1.

Planes: Parametric format: 𝑟 = 𝑎 + 𝜆𝑏 + 𝜇𝑐

By multiplying by 𝑛, perpendicular to r (i.e. b and c) we have 𝑟. 𝑛 = 𝑎. 𝑛 + 𝜆𝑏. 𝑛 + 𝜇𝑐. 𝑛

⇒ 𝑟. 𝑛 = 𝑎. 𝑛 = 𝑑 the Cartesian format

To find the angle between a line and a plane, find the angle between the normal (observed

from Cartesian or derived from Parametric) and the line.

The shortest distance from point P to line AB is 𝐴𝑃 ×𝐴𝐵

𝐴𝐵 , A and B are two points on the line.

The shortest distance from two lines is found by: Finding vector 𝐴𝐵 from a point on one line

to a point on the other, finding vector 𝑛 perpendicular to both line, then calculating 𝐴𝐵 ∙𝑛

𝑛 .

Inverse Matrices

If A = 𝑎 𝑏𝑐 𝑑

, then A-1 = 1

𝑎𝑑−𝑏𝑐

𝑑 −𝑏−𝑐 𝑎

⇒ 𝑀−1

Any matrix with no inverse is called a singular matrix.

For a 3 by 3 matrix:

1- Find the matrix of minor determinants.

2- Alter the sign in this pattern + − +− + −+ − +

3- Transpose and divide by 𝑀

(𝐴𝐵𝐶 … )−1 = …𝐶−1𝐵−1𝐴−1

Linear Equations

The planes can either:

meet at a triangular prism with no solutions

form a sheaf (with a common line of solutions)

or intersect at a unique point.

The matrix equation will be of the form 𝐴𝑟 = 𝑏, then 𝐴−1 𝐴𝑟 = 𝐴−1 ⇒ 𝑟 = 𝐴−1𝑏

If vectors are combinations of other vectors, they are linearly dependent.

The best way to test is to examine for

𝑎1

𝑎2

𝑎3

,

𝑏1

𝑏2

𝑏3

&

𝑐1

𝑐2

𝑐3

, if 𝑑𝑒𝑡

𝑎1 𝑏1 𝑐1

𝑎2 𝑏2 𝑐2

𝑎3 𝑏3 𝑐3

= 0 then A,B & C

are linearly dependent.

Invariant Lines and the more general case; eigenvectors

Invariant points are points left unchanged after transformation 𝑀. Found by solving 𝑀𝑥 = 𝑥

Invariant lines are lines of which the points stay on the line after transformation 𝑀. Found by

solving 𝑀𝑥

𝑚𝑥= 𝑥′

𝑚𝑥′ . All parallel lines are also invariant lines. E.g.

0 11 0

𝑥

𝑚𝑥 + 𝑐 which

makes 𝑚𝑥 + 𝑐

𝑥 so 𝑥 = 𝑚(𝑚𝑥 + 𝑐) + 𝑐 for all x.

If Matrix and vector satisfy 𝑀𝑣 = 𝜆𝑣, the 𝑣 is an eigenvector and 𝜆 an eigenvalue.

The eigenvectors of 𝑀 determines the direction of all invariant lines through the origin.

𝑀𝑣 = 𝜆𝑣 ⇒ 𝑀 − 𝜆𝐼 𝑣 = 0 If 𝑀 − 𝜆𝐼 has an inverse then 𝑣 = 0 but 𝑣 ≠ 0 hence:

𝑀 − 𝜆𝐼 = 0 can be used to determine the eigenvalues, 𝜆, of 𝑀. And hence eigenvectors

can be found from 𝑀 − 𝜆𝐼 = 0

Matrix 𝑀 satisfies 𝑀 = 𝑉𝐷𝑉−1 where 𝑉 = 𝑉𝑖1 𝑉𝑖2

𝑉𝑗1 𝑉𝑗2 and D =

𝜆1 00 𝜆2

If 𝑀 = 𝑉𝐷𝑉−1, then 𝑀𝑛 = 𝑉𝐷𝑛𝑉−1