Chapter 5 - Matrices 1

Question 1

Describe a matrix of an order n x m.

Answer 1

Question 2

How do determine the order of a matrix?

Answer 2

A matrix with n rows and m columns has order n x m.

Question 3

What is possible on two matrices that have the same order?

Answer 3

If two matrices have the same order then they can be added or subtracted by adding or subtracting their corresponding elements.

Question 4

What is the rule regarding matrices being multiplied by constants?

Answer 4

To multiply a matrix by a constant, multiply each of its elements by that constant.

Question 5

What is the zero matrix?

Answer 5

The zero matrix is a matrix of any order with all elements equal to zero.

Question 6

If two matrices can be multiplied, what are those matrices known to be able to do?

Answer 6

If two matrices can be multiplied, then it is said that they are conformable for multiplication.

Question 7

What is the order of the product of matrices n x m, and m x p?

Answer 7

The product of matrices n x m and m x p has order n x p.

Question 8

What is the rule for squaring matrices?

Answer 8

It is only possible to square a matrix if it is a square matrix, meaning the matrix has the same number of orders as rows.

Question 9

How do you transpose a matrix?

Answer 9

You transpose a matrix by swapping the rows and columns.

Question 10

How do you find the image of a vector?

Answer 10

In order to find the image of a vector (x,y) under a transformation T = (a b, c d) (2x2), we premultiply the vector by the matrix. (T x vector)

Question 11

How do you represent a reflection in the line y = x?

Answer 11

The matrix

( 0 1 )
( 1 0 )

represents a reflection in the line y = -x.

Question 12

How do you represent a reflection in the line y = -x?

Answer 12

The matrix

( 0 -1 )
( -1 0 )

represents a reflection in the line y = -x.

Question 13

What is the identity matrix?

Answer 13

The matrix, I =

( 1 0 )
( 0 1 )

is the identity matrix, which works on all 2x2 matrices. Multiplying this transformation has no effect on the original vector.

Question 14

What matrix represents a stretch of scale factor k parallel to the x-axis?

Answer 14

The matrix

( k 0 )
( 0 1 )

represents a stretch of scale factor k parallel to the x-axis.

Question 15

What matrix represents a stretch of scale factor k parallel to the y-axis?

Answer 15

The matrix

(1 0)
(0 k)

represents a stretch of scale factor k parallel to the y-axis.

Question 16

What matrix represents an enlargement of scale factor k centre the origin?

Answer 16

The matrix

(k 0)
(0 k)

represents an enlargement of scale factor k centre the origin.

Question 17

How do you represent an anticlockwise rotation by angle theta about the origin?

Answer 17

The matrix

(cos(θ) -sin(θ))
(sin(θ) cos(θ))

represents an anticlockwise rotation by angle (theta) about the origin.

Question 18

For 3D reflections, what represents a reflection in x = 0?

Answer 18

The matrix

(-1 0 0)
(0 1 0)
(0 0 1)

represents a reflection in x = 0.

Question 19

For 3D reflections, what represents a reflection in y = 0?

Answer 19

The matrix

(1 0 0)
(0 -1 0)
(0 0 1)

represents a reflection in y = 0.

Question 20

For 3D reflections, what represents a reflection in z = 0?

Answer 20

The matrix

(1 0 0)
(0 1 0)
(0 0 -1)

represents a reflection in z = 0.

Question 21

For 3D rotations, what represents a rotation around x=0?

Answer 21

The matrix

(1 0 0)
(0 cos(θ) -sin(θ))
(0 sin(θ) cos(θ))

represents a rotation around the x axis.

Question 22

For 3D rotations, what represents a rotation around y=0?

Answer 22

The matrix

(cos(θ) 0 -sin(θ))
(0 1 0)
(-sin(θ) 0 cos(θ))

represents a rotation around the y-axis.

Question 23

For 3D rotations, what represents a rotation around z=0?

Answer 23

The matrix

(cos(θ) -sin(θ) 0)
(sin(θ) cos(θ) 0)
(0 0 1)

represents a rotation around the z-axis.

Question 24

What is an invariant point?

Answer 24

A point which is unaffected by a transformation is known as an invariant point.

Given a transformation matrix T and position vector x, if Tx = x, then x represents an invariant point.

Question 25

What is the rule for linear transformations?

Answer 25

For any linear transformation, (0,0) is an invariant point.

Question 26

If every point on a line is mapped to itself under a transformation then what is it known as?

Answer 26

If every point on a line is mapped to itself under a transformation then it is known as a line of invariant points.

Question 27

If every point on a line is mapped to another point one the same line then, what is it known as?

Answer 27

If every point on a line is mapped to another point one the same line then, it is known as an invariant line.

Question 28

What is the determinant of a matrix?

Answer 28

If A = (a b) (c d) then the determinant of a is ad - bc and is denoted |A| or det(A)

Question 29

What defines a singular matrix?

Answer 29

If det(A) = 0, then A is called a singular matrix.

Question 30

How do you find the inverse matrix?

Answer 30

If A = (a b) (c d) then the inverse matrix is given by A⁻¹ = 1/det(A) times... (d -b) (-c a)

Question 31

What is the rule regarding any square, non-singular matrix A?

Answer 31

For any square, non-singular matrix A: det(A⁻¹) = 1/(det(A))

Question 32

Define an Eigenvector? Therefore, define an Eigenvalue?

Answer 32

A vector matrix that is scaled linearly by a non-traditional transformatory matrix is called an Eigenvector, and the scale factor is known as the Eigenvalue.