Matrices

Question 1

multiplying by the identity matrix

Answer 1

multiplying by 1

Question 2

multiplying by the null/zero matrix

Answer 2

multiplying by 0

Question 3

order/dimension of a matrix

Answer 3

the number of rows/columns

Question 4

you can only add matrices if they have…

Answer 4

the same order

Question 5

AB != BA

Answer 5

matrix multiplication is not commutative

Question 6

a(bc) = (ab)c

Answer 6

Associative Property of Multiplication

Question 7

a(b+c) = ab + ac

Answer 7

Distributive

Question 8

conformable

Answer 8

mxn order can be multiplied by nxp order giving a matrix of order mxp

Question 9

a stretch parallel to the x-axis with scale factor k

Answer 9

K 0

0 1

Question 10

A stretch parallel to the y-axis with scale factor k

Answer 10

1 0

0 K

Question 11

A enlargement centre of origin with scale factor k

Answer 11

K 0

0 K

Question 12

A rotation 90 anti-clockwise about the origin

Answer 12

0 -1

1 0

Question 13

A rotation 90 clockwise about the origin

Answer 13

0 1

-1 0

Question 14

leading diagonal rule rotation matrices

Answer 14

the elements in the leading diagonal are the exact same

the elements in the opposite diagonal are the same but with opposite signs

Question 15

A rotation of any angle anticlockwise

Answer 15

cosθ -sinθ

sinθ cosθ

Question 16

reflection in the x-axis

Answer 16

1 0

0 -1

Question 17

reflection in the y-axis

Answer 17

-1 0

0 1

Question 18

reflection in the line y=x

Answer 18

0 1

1 0

Question 19

reflection in the line y=-x

Answer 19

0 -1

-1 0

Question 20

Shear

Answer 20

a transformation in which all the point are translated parallel to a particular line by a factor which is proportional to the distance of the point from a shear line

Question 21

shear parallel to the x-axis

Answer 21

1 K

0 1

Question 22

shear parallel to the y-axis

Answer 22

1 0

K 1

Question 23

reflection in the yz plane

Answer 23

-1 0 0
0 1 0
0 0 1

Question 24

reflection in the xz plane

Answer 24

1 0 0
0 -1 0
0 0 1

Question 25

rotation of 180 about the z axis

Answer 25

-1 0 0 0 -1 0 0 0 1

Question 26

Invarient Point

Answer 26

a point which is mapped to itself by the transformation

Question 27

How to find a line of invarient points

Answer 27

1. multiple the matric by x y and make it equal to x y 2. multiply out into two equations (top and bottom) 3. if (ax + by = x)= (cx +dy = y) then all the points along ax+by = x are invarient points 4. else (0,0) is the only invarient point

Question 28

How to find an invarient line

Answer 28

1. multiply the matrix by x y and set it equal to x' y' 2. replace y with mx+c 3. replace y' = mx' + c 4. rearrange and factorise into two parts: the x's and the c's 5. find the value of m to make the x bracket 0 6. sub the value into c's brack 7. IF != 0 then c=0, sub values back into 4. these are the invairent lines 8. IF = 0 then c is some contant k, sub these values back into 4, these are invarient linez

Question 29

determinant

Answer 29

the scale factor of the transformation

Question 30

determinant of a 2x2

Answer 30

ad-bc

Question 31

if the determinant is zero the matrix is...

Answer 31

singular

Question 32

Inverse of a 2x2

Answer 32

1/determinant * d -b | -c a

Question 33

AA^-1 =

Answer 33

identity matrix

Question 34

(AB)^-1 =

Answer 34

A^-1 B^-1

Question 35

determinant of a 3x3

Answer 35

expand a column / row | e.g, multiply each of the elements in a row by its cofactor

Question 36

inverse of a 3x3 matrix

Answer 36

- find the determinant - find the cofactor of each element - replace each element with its cofactor - reflect in the diagonal to get the adjugate/adjoint matrix (take the transpose) - divide by the determinant

Question 37

solving simultaneous equations

Answer 37

- separate out the simultaneous equations into a matrix of the coefficiants multiplied by the unknowns = the constants - multiply the constants by the inverse of the matrix

Question 38

what type of solution are there for matrix simultaneous equations?

Answer 38

- no solution (determinant = 0) - unique solution - infinitely many solutions

Question 39

geometric representation of unique solution

Answer 39

planes intersect at a single point

Question 40

geometric representation of infinitely many solutions

Answer 40

sheaf of planes

Question 41

geometric representation of no solutions

Answer 41

- three parallel planes - two planes are parallel and the third is not - triangular prism

Question 42

equations for parallel planes

Answer 42

coefficients are the same but constants are different

Question 43

1 point of intersection (3 simultaneous equations, 3 planes)

Answer 43

detM != 0, non-singular solutions can be found

Question 44

3 parallel planes

Answer 44

detM = 0, singular, all three equations are coincident

Question 45

2 parallel planes

Answer 45

detM = 0, singular, two of the equations are coincident

Question 46

3 equations the same/ multiples (coincident)

Answer 46

3 parallel planes

Question 47

2 equations same/ multiples (coincident)

Answer 47

2 parallel planes

Question 48

Sheaf

Answer 48

detM = 0, singular, equations consistent

Question 49

What is a sheaf

Answer 49

where all three planes cross but instead of a single point of intersection there is a central line down all three points

Question 50

equations consistent

Answer 50

sheaf (infinitely many solutions)

Question 51

equations not consistent and not coincident

Answer 51

triangular prism

Question 52

consistent equations

Answer 52

They can be solved but for infinite solutions, multiples of each other

Question 53

example of consistent equations

Answer 53

x + y = 1 | 2x + 2y = 2

Question 54

example of inconsistent equations

Answer 54

5x - y = 17 | 5x - y = 15

Question 55

reflection in y axis function form

Answer 55

f(-x)

Question 56

reflection in x axis function form

Answer 56

-f(x)