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AQA Mathematics > Matrices > Flashcards

Matrices Flashcards

1
Q

multiplying by the identity matrix

A

multiplying by 1

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2
Q

multiplying by the null/zero matrix

A

multiplying by 0

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3
Q

order/dimension of a matrix

A

the number of rows/columns

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4
Q

you can only add matrices if they have…

A

the same order

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5
Q

AB != BA

A

matrix multiplication is not commutative

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6
Q

a(bc) = (ab)c

A

Associative Property of Multiplication

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7
Q

a(b+c) = ab + ac

A

Distributive

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8
Q

conformable

A

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

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9
Q

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

A

K 0

0 1

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10
Q

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

A

1 0

0 K

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11
Q

A enlargement centre of origin with scale factor k

A

K 0

0 K

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12
Q

A rotation 90 anti-clockwise about the origin

A

0 -1

1 0

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13
Q

A rotation 90 clockwise about the origin

A

0 1

-1 0

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14
Q

leading diagonal rule rotation matrices

A

the elements in the leading diagonal are the exact same

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

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15
Q

A rotation of any angle anticlockwise

A

cosθ -sinθ

sinθ cosθ

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16
Q

reflection in the x-axis

A

1 0

0 -1

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17
Q

reflection in the y-axis

A

-1 0

0 1

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18
Q

reflection in the line y=x

A

0 1

1 0

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19
Q

reflection in the line y=-x

A

0 -1

-1 0

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20
Q

Shear

A

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

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21
Q

shear parallel to the x-axis

A

1 K

0 1

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22
Q

shear parallel to the y-axis

A

1 0

K 1

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23
Q

reflection in the yz plane

A

-1 0 0
0 1 0
0 0 1

24
Q

reflection in the xz plane

A

1 0 0
0 -1 0
0 0 1

25
Q

rotation of 180 about the z axis

A

-1 0 0
0 -1 0
0 0 1

26
Q

Invarient Point

A

a point which is mapped to itself by the transformation

27
Q

How to find a line of invarient points

A
  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
28
Q

How to find an invarient line

A
  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
29
Q

determinant

A

the scale factor of the transformation

30
Q

determinant of a 2x2

A

ad-bc

31
Q

if the determinant is zero the matrix is…

A

singular

32
Q

Inverse of a 2x2

A

1/determinant * d -b

-c a

33
Q

AA^-1 =

A

identity matrix

34
Q

(AB)^-1 =

A

A^-1 B^-1

35
Q

determinant of a 3x3

A

expand a column / row

e.g, multiply each of the elements in a row by its cofactor

36
Q

inverse of a 3x3 matrix

A
  • 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
37
Q

solving simultaneous equations

A
  • 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
38
Q

what type of solution are there for matrix simultaneous equations?

A
  • no solution (determinant = 0)
  • unique solution
  • infinitely many solutions
39
Q

geometric representation of unique solution

A

planes intersect at a single point

40
Q

geometric representation of infinitely many solutions

A

sheaf of planes

41
Q

geometric representation of no solutions

A
  • three parallel planes
  • two planes are parallel and the third is not
  • triangular prism
42
Q

equations for parallel planes

A

coefficients are the same but constants are different

43
Q

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

A

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

44
Q

3 parallel planes

A

detM = 0, singular, all three equations are coincident

45
Q

2 parallel planes

A

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

46
Q

3 equations the same/ multiples (coincident)

A

3 parallel planes

47
Q

2 equations same/ multiples (coincident)

A

2 parallel planes

48
Q

Sheaf

A

detM = 0, singular, equations consistent

49
Q

What is a sheaf

A

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

50
Q

equations consistent

A

sheaf (infinitely many solutions)

51
Q

equations not consistent and not coincident

A

triangular prism

52
Q

consistent equations

A

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

53
Q

example of consistent equations

A

x + y = 1

2x + 2y = 2

54
Q

example of inconsistent equations

A

5x - y = 17

5x - y = 15

55
Q

reflection in y axis function form

A

f(-x)

56
Q

reflection in x axis function form

A

-f(x)

AQA Mathematics (68 decks)

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