Matrices

Question 1

Non-commutativity meaning

Answer 1

AB != BA

Question 2

Associativity meaning

Answer 2

(AB)C = A(BC)

Question 3

Identity meaning

Answer 3

A = AI = IA

Question 4

Matrices Inverse definition

Answer 4

AA^-1 = I = A^-1 A

Question 5

Product of inverses, (AB)^-1 =

Answer 5

B^-1 A^-1

Question 6

Determinant of a 2x2 matrix

Answer 6

ad-bc

Question 7

Determinant of a 3x3 matrix

Answer 7

a det(efhi) - b det(dfgi) + c det(degh)

Question 8

If det(M) = 0,

Answer 8

M is singular and has no inverse

Question 9

Matrix of minors

Answer 9

For every element, draw cross through and find determinant

Question 10

Matrix of cofactors

Answer 10

Cross +, diamond -, apply to matrix of minors

Question 11

Transpose of a matrix

Answer 11

Rows and columns swap

Question 12

2x2 inverse

Answer 12

1/det (d -b
-c a)

Question 13

3x3 inverse

Answer 13

1/det C^T where C is the matrix of cofactors

Question 14

Solve matrix equations

Answer 14

Write as (matrix)(x y z) = (answers)
Do inverse to get x y z

Question 15

Work out what matrix plane equations look like

Answer 15

Check det M, if it isn’t 0 the planes meet at 1 point.
If it is and is consistent, multiple solutions exist like a sheaf
If it is inconsistent and isn’t parallel, it forms a prism

Question 16

How to work out invariant points

Answer 16

M(x y) = (x y)

Question 17

How to work out invariant lines, y = mx + c

Answer 17

M(x mx+c) = (x’ y’)
sub back into y = mx + c
Solve

Question 18

How to work out invariant lines, y = ax

Answer 18

M(x ax) = k(x ax)

Question 19

What is the area scale factor?

Answer 19

det M

Question 20

Successive transformations

Answer 20

PQ is Q then P

Question 21

Rotation x anticlockwise

Answer 21

( cosx -sinx
sinx cosx)

Question 22

Reflection in plane

Answer 22

3 x 3 identity matrix, turn corresponding column negative

Question 23

Rotation x anticlockwise about x axis.

Answer 23

1 0 0
0 cosx -sinx
0 sinx cosx

Question 24

Rotation x anticlockwise about y axis.

Answer 24

cosx 0 sinx
0 1 0
-sinx 0 cosx

Question 25

Rotation x anticlockwise about z axis

Answer 25

cosx -sinx 0
sinx cosx 0
0 0 1

Question 26

To undo transformations

Answer 26

P^-1(PQ)Q^-1

Question 27

Proof by Induction Steps

Answer 27

Base case, n = 1
Assumption, n = k
Inductive case, n = k+1
Conclude true for all n £ N

Question 28

Divisibility Induction

Answer 28

Use f(k+1) - f(k)