matrices and matrix methods

Question 1

How would a 2-dimensional image be typically be represented?

Answer 1

256 x 256 matrix

Question 2

What is a matrix?

Answer 2

A set of real or complex numbers arranged in rows + columns to form a rectangular array

Question 3

What is m x n matrix?

Answer 3

A matrix having m rows and n columns

Question 4

What does row matrix consist of?

Answer 4

1 row only

Question 5

What do column matrix consist of?

Answer 5

1 column only

Question 6

What does square matrix have?

Answer 6

Equal number of rows and columns

Question 7

What do each element in a matrix have?

Answer 7

Its own ‘address’ or ‘location’ defined by a systems of double suffixes

Question 8

What do the first and second suffix indicate?

Answer 8

First = row
Second = column

Question 9

How can a 3D matrix be written as?

Answer 9

(aijk)

Question 10

How can a vector be represented as?

Answer 10

A matrix of order 1 x n

Question 11

How can matrices be added or subtracted?

Answer 11

2 matrices must be of the same order

Question 12

What is the application of addition and substraction of matrices?

Answer 12

Imaging application

Adding multiple averages to increase SNR

Question 13

How can matrixes be multiplied?

Answer 13

By single number (i.e. scalar)

Each individual element is multiplied by that factor

Question 14

What is the application of multiplication of matrices?

Answer 14

Changing image display brightness

Question 15

What is equation for multiplication of matrices?

Answer 15

k(aij) = (kaij)

Question 16

When is matrix multiplication only possible?

Answer 16

When no. of column in first is equal to the number of rows in second

a.b = c

Question 17

When are transpose of a matrix formed?

Answer 17

When the rows and columns of matrix are interchanged

• The first row becomes the column
• The second row becomes the second column
• The third row becomes the third column

Question 18

What is a square matrix a order of?

Answer 18

m x m

Question 19

What is a square matrix?

Answer 19

symmetric
symmetric about leading diagonal

if aij=aji

Question 20

What is a diagonal matrix?

Answer 20

A square matrix with all elements 0 except those on the leading diagonal

Question 21

What is unit matrix?

Answer 21

A diagonal matrix in which elements along the leading diagonal are unity

Question 22

What is the application of special matrices?

Answer 22

Imaging application:

symmetric and diagonal matrices feature in DTI theory

Question 23

What is the unit matrix denoted by and what it is its role?

Answer 23

Denoted by I

I has the same function as the number 1 in standard scalar algebra

Question 24

What is the consequence of multiplying matrix by identity matrix I?

Answer 24

Leaves it unchanged

Question 25

What is determinant of a matrix?

Answer 25

A value associated with square matrix

Computed from the entries of the matrix by a specific arithmetic expression (or using computer function)

Question 26

What do determinant of matrix provide information on?

Answer 26

Useful in matrix representation of linear equations

Useful when matrix represents a geometric transformation

Question 27

How is determinat of 2 x 2 matrix written symbollically as?

Answer 27

det A or magnitude of A
[A B C D] =
ad-bc

Question 28

What is the determinant of a 3x3 matrix?

Answer 28

(aei+bfg+cdh)-(ceg+afh+bdi)

Question 29

What is the determinant?

Answer 29

Just a number
It gives useful information about the matrix
It is needed to compute the inverse of the matrix

Question 30

What is matrix division?

Answer 30

Inverse of matrix

Only applies to square matrices

Question 31

What is the important application of a matrix division?

Answer 31

Image transformation
Image analysis
Performed in software

Question 32

How is the inverse of a 2x2 matrix determined by?

Answer 32

Rearranging the matrix elements and dividing by the determinant

Question 33

Inverse of 2x2 matrix?

Answer 33

The elements on the leading diagonal are swapped, the remaining elements are multiplied by -1, and the resulting matrix divided by the determinant of A