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Matrices > Matrices Year 1 > Flashcards

Matrices Year 1 Flashcards

1
Q

What is the order of the matrix?

A

Rows x Columns

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2
Q

What is a square matrix?

A

Same number of row as columns

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3
Q

What is the identity matrix?

A

1 0

0 1

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4
Q

What is the null matrix?

A

0 0

0 0

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5
Q

What is the transpose of a matrix?

A

Writing the columns as rows and vice versa

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6
Q

When can you add and subtract a matrix?

A

When they have the same order- add and subtract each element

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7
Q

Is matrix multiplication associative?

A

Yes

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8
Q

Is matrix multiplication commutative?

A

No

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9
Q

Define object

A

Original point/shape

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10
Q

Define image

A

The transformed shape

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11
Q

Define mapping

A

The transformation

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12
Q

Define i : 1

0

A

Unit vector in x axis (2D)

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13
Q

What is a unit vector?

A

A vector with a magnitude of one

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14
Q

Define j : 0

1

A

Unit vector in y axis (2D)

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15
Q

Define i: 1
0
0

A

Unit vector in x direction (3D)

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16
Q

Define j: 0
1
0

A

Unit vector in y direction (3D)

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17
Q

Define k: 0
0
1

A

Unit vector in z direction (3D)

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18
Q

How would you be able to identify the type of transformation?

e.g what is 2 0
0 2

A

Draw the transformation on a graph
1) Find the new coordinates by multiplying transformation individually by (1,0) (1,1) and (0,1) matrices

e.g

2 0 X 1 = new (1,0) point
0 2 0

2 0 X 1 = new (1,1) point
0 2 1

2 0 X 0 = new (0,1) point
0 2 1

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19
Q

What are the coordinates of the unit square in 2D?

A

(0,1)
(1,0)
(1,1)

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20
Q

Give the matrix for a enlargement (2D)

A

a 0
0 a

Where a > 0

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21
Q

Give the matrix of a stretch parallel to x axis (2D)

A

a 0

0 1

22
Q

Give the matrix of a stretch parallel to y axis (2D)

A

1 0

0 a

23
Q

Give the matrix of a shear parallel to x axis and how would you describe it? (2D)

A

1 a
0 1

description: x axis is invariant and list a point that has changed ie a point not on the x axis. Give its original position and new transformed position e.g (0,1) has been mapped to (a,1)

24
Q

Give the matrix of a shear parallel to y axis and how would you describe it? (2D)

A

1 0
a 1

description: y axis is invariant and list a point that has changed ie a point not on the y axis. Give its original position and new transformed position. e.g (1,0) has been mapped to (1,a)

25
Q

How do you find the shear factor?

A

distance between original point and invariant axis

26
Q

Give the matrix of a reflection in the x axis (2D)

A

1 0

0 -1

27
Q

Give the matrix of a reflection in the y axis (2D)

A

-1 0

0 1

28
Q

Give the matrix of a reflection in the line y=x (2D)

A

0 1

1 0

29
Q

Give the matrix of a reflection in the line y=-x (2D)

A

0 -1

-1 0

30
Q

Give the matrix of a reflection in the line x= 0 aka ZY plane (3D)

A

-1 0 0
0 1 0
0 0 1

31
Q

Give the unit matrix for a 3D shape

A

1 0 0
0 1 0
0 0 1

32
Q

Give the matrix of a reflection in the line y= 0 aka XZ plane (3D)

A

1 0 0
0 -1 0
0 0 1

33
Q

Give the matrix of a reflection in the line z= 0 aka XY plane (3D)

A

1 0 0
0 1 0
0 0 -1

34
Q

Give the matrix for the anticlockwise rotation about the origin through angle θ (given in formula sheet) (2D)

A

cosθ -sinθ

sinθ cosθ

35
Q

What is the order of successive transformations such AB for example?

A

AB (matrix A X matrix B) means do B first then A

36
Q

What is an invariant point?

A

A point that maps to itself under the transformation

a b X x = x
c d y y

37
Q

How would you find the line of invariant points?

e.g for the matrix

2 -1
1 0

A

Multiply matrix by (x,y) matrix and make it equal to (x,y)

e.g

2 -1 X x = x
1 0 y y

so..

2x - y = x
x y

so line is x = y

38
Q

What is an invariant line?

A

Every point on the invariant line maps to either itself or another point on the line

39
Q

How do you find an invariant line?

e.g find the invariant line on

5 1
2 4

A

1) multiply matrix by x,y

5 1 X x = x’
2 4 y y’

2) equate x’ and y’

so x’ = 5x+y and y’ = 2x+4y

3) remove y by subbing y = mx+x

so x’ = 5x + mx + c y’ = 2x + 4(mx+c)
x’ = x(5+m) + c y’ = x(2+4m) + 4c

4) Put x’ and y’ into y’ = mx’ +c

x(2+4m) + 4c = m [x(5+m) + c] + c

5) expand and equate x and c to form y= mx +c equation

so 0 = (m^2 + m + 2)x + (m-3)c

for this to equal 0 the x and c must be zero
so for x: m^2 + m - 2 = 0 so m= 1 or -2
for c: m-3 = 0 so m = 3 or c = 0

so m= 1,-2 and c = 0
m can not equal 3 or else x coefficient will not equal zero

6) sub in m and c into y= mx +c

so final equations of invariant lines are
y=x and y=-2x

40
Q

How do you find the determinant of a 2x2 matrix?

a b
c d

A

ad-bc

41
Q

What does the determinant represent in a transformation?

A

The area scale factor

42
Q

What does it mean in the transformation if the determinant is negative?

A

The orientation of the vertices is reversed

43
Q

What does |M| mean?

A

The determinant of matrix m

44
Q

|MN| =

A

|M| x |N|

45
Q

If the determinant is zero what does this mean about the transformation? and how would you find that line?

A

It maps the object to a line - find the line by looking at the relationship the new x and y values have:

6 4 x p = 6p +4q
3 2 q = 3p+2q

so here the object is mapped to line 2y=x

46
Q

How do you find the inverse of a 2x2 matrix?

a b
c d

A

1 d -b
——– x
ad-bc -c a

47
Q

What does it mean if the determinant is zero for its inverse?

A

It does not have an inverse - it is singular.

48
Q

(MN) -1 =

A

M-1 x N-1

49
Q

How do you find the determinant of a 3x3 matrix?

A

(a1 x matrix of minor when removing a1 ) - (a2 x matrix of minor when removing a2) + (a3 x matrix of minor when removing a3)

50
Q

How do you find the inverse of a 3x3 matrix?

A

1) find matrix of minors
2) put in signs
3) transpose
4) multiply by 1/determinant

51
Q

How do you solve simultaneous equations using matrices?

A

1) convert simultaneous equations into a matrix multiplication

(2x2 matrix of coefficients called matrix M) x (2x1 matrix of x and y) = (2x1 matrix of answers)

2) Premultiply both sides by inverse of M to find x and y matrix

52
Q

What are the 3 cases for matrix geometric representation of 2 simultaneous lines?

A

CASE 1 - det M doesn’t equal 0
One solution

CASE 2- det M equals 0
lines are parallel so no solution

CASE 3- det M equals 0
lines are the same so coincident so infinite solutions (matrices are scalar multiples of each other)

Matrices (2 decks)

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