Matrices Year 1

Question 1

What is the order of the matrix?

Answer 1

Rows x Columns

Question 2

What is a square matrix?

Answer 2

Same number of row as columns

Question 3

What is the identity matrix?

Answer 3

1 0

0 1

Question 4

What is the null matrix?

Answer 4

0 0

0 0

Question 5

What is the transpose of a matrix?

Answer 5

Writing the columns as rows and vice versa

Question 6

When can you add and subtract a matrix?

Answer 6

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

Question 7

Is matrix multiplication associative?

Answer 7

Yes

Question 8

Is matrix multiplication commutative?

Answer 8

No

Question 9

Define object

Answer 9

Original point/shape

Question 10

Define image

Answer 10

The transformed shape

Question 11

Define mapping

Answer 11

The transformation

Question 12

Define i : 1

0

Answer 12

Unit vector in x axis (2D)

Question 13

What is a unit vector?

Answer 13

A vector with a magnitude of one

Question 14

Define j : 0

1

Answer 14

Unit vector in y axis (2D)

Question 15

Define i: 1
0
0

Answer 15

Unit vector in x direction (3D)

Question 16

Define j: 0
1
0

Answer 16

Unit vector in y direction (3D)

Question 17

Define k: 0
0
1

Answer 17

Unit vector in z direction (3D)

Question 18

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

e.g what is 2 0
0 2

Answer 18

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

Question 19

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

Answer 19

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

Question 20

Give the matrix for a enlargement (2D)

Answer 20

a 0
0 a

Where a > 0

Question 21

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

Answer 21

a 0

0 1

Question 22

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

Answer 22

1 0

0 a

Question 23

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

Answer 23

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)

Question 24

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

Answer 24

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)

Question 25

How do you find the shear factor?

Answer 25

distance between a point and its image ---------------------------------------------------------- distance between original point and invariant axis

Question 26

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

Answer 26

1 0 | 0 -1

Question 27

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

Answer 27

-1 0 | 0 1

Question 28

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

Answer 28

0 1 | 1 0

Question 29

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

Answer 29

0 -1 | -1 0

Question 30

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

Answer 30

-1 0 0 0 1 0 0 0 1

Question 31

Give the unit matrix for a 3D shape

Answer 31

1 0 0 0 1 0 0 0 1

Question 32

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

Answer 32

1 0 0 0 -1 0 0 0 1

Question 33

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

Answer 33

1 0 0 0 1 0 0 0 -1

Question 34

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

Answer 34

cosθ -sinθ | sinθ cosθ

Question 35

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

Answer 35

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

Question 36

What is an invariant point?

Answer 36

A point that maps to itself under the transformation a b X x = x c d y y

Question 37

How would you find the line of invariant points? e.g for the matrix 2 -1 1 0

Answer 37

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

Question 38

What is an invariant line?

Answer 38

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

Question 39

How do you find an invariant line? e.g find the invariant line on 5 1 2 4

Answer 39

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

Question 40

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

Answer 40

ad-bc

Question 41

What does the determinant represent in a transformation?

Answer 41

The area scale factor

Question 42

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

Answer 42

The orientation of the vertices is reversed

Question 43

What does |M| mean?

Answer 43

The determinant of matrix m

Question 44

|MN| =

Answer 44

|M| x |N|

Question 45

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

Answer 45

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

Question 46

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

Answer 46

1 d -b -------- x ad-bc -c a

Question 47

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

Answer 47

It does not have an inverse - it is singular.

Question 48

(MN) -1 =

Answer 48

M-1 x N-1

Question 49

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

Answer 49

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

Question 50

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

Answer 50

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

Question 51

How do you solve simultaneous equations using matrices?

Answer 51

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

Question 52

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

Answer 52

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)