Matrix algebra

Question 1

What are the dimensions of a matrix?

Answer 1

The number of rows by the number of columns

Question 2

What property must two matrices have for addition to be possible?

Answer 2

They must have the same dimensions

Question 3

How can two matrices A and B be added?

Answer 3

Each element in matrix A is simply added to the corresponding value in matrix B

Question 4

How can a matrix be multiplied by a scalar?

Answer 4

By multiplying each element by the scalar

Question 5

What property must two matrices A and B have for multiplication to be possible?

Answer 5

The number or columns in A must be equal to the number of rows in B

Question 6

If matrix A has dimensions a×b and matrix B has dimensions b×c, what are the dimensions of matrix AB?

Answer 6

a×c

Question 7

How can two matrices A and B be multiplied?

Answer 7

First work out the dimensions of the product. Then to find the element in the mth row and the nth column in the matrix AB: go through the mth row of A and, for each element in it, multiply it by the corresponding value in the nth column of B, and then add up these products

Question 8

What can be said about the commutativity of matrix multiplication?

Answer 8

It is not commutative

Question 9

What can be said about the associativity of matrix multiplication?

Answer 9

It is associative

Question 10

What is a transformation?

Answer 10

An operation that moves all points in a plane according to some rule

Question 11

What is a linear transformation?

Answer 11

A transformation that only involves linear expressions of x and y, and also leaves the origin unmoved

Question 12

What is a position vector?

Answer 12

The vector from the origin to a point

Question 13

What is an image?

Answer 13

The point to which a position vector is moved to after a transformation

Question 14

What can be said about the distributivity of matrix multipliation?

Answer 14

It is distributive

Question 15

For a linear transformation T:[(x),(y)]→[(ax+by),(cx+dy)], how can T be represented in matrix form?

Answer 15

[(a,b),(c,d)]

Question 16

Given that a shape S has coordinates (x₁,y₁), (x₂,y₂), …, (xᵣ,yᵣ), how can the coordinates of the image of S under a transformation matrix M be found?

Answer 16

By transforming the coordinates into column vectors, making a matrix out of these vectors, and then multiplying M by this matrix, i.e. M[(x₁,x₂,…,xᵣ),(y₁,y₂,…,yᵣ)]

Question 17

How may the transformation represented by a matrix be found?

Answer 17

By applying it to the vectors [(1),(0)] and [(0),(1)] and observing the ensuing transformations

Question 18

What matrix represents an anticlockwise rotation of θ around the origin?

Answer 18

[(cos(θ),-sin(θ)),(sin(θ),cos(θ)]

Question 19

What matrix represents a reflection in the line x=0?

Answer 19

[(-1,0),(0,1)]

Question 20

What matrix represents a reflection in the line y=0?

Answer 20

[(1,0),(0,-1)]

Question 21

What matrix represents a reflection in the line y=x?

Answer 21

[(0,1),(1,0)]

Question 22

What matrix represents a reflection in the line y=-x?

Answer 22

[(0,-1),(-1,0)]

Question 23

What matrix represents an enlargement of scale factor n around the origin?

Answer 23

[(n,0),(0,n)]

Question 24

Given matrices A and B representing two transformations, how can the matrix for the transformation of A followed by the transformation of B be calculated?

Answer 24

BA

Question 25

What is the identity matrix?

Answer 25

A square matrix with ones down the main diagonal and zeros elsewhere

Question 26

How is the identity matrix notated?

Answer 26

I

Question 27

What property does the identity matrix have?

Answer 27

For any matrix A, AI=IA=A

Question 28

For a matrix [(a,b),(c,d)], how can the determinant be calculated?

Answer 28

ad-bc

Question 29

How is the determinant of a matrix M notated?

Answer 29

det(M)

Question 30

What is the inverse of a matrix M?

Answer 30

The matrix M⁻¹ such that MM⁻¹=M⁻¹M=I

Question 31

How is the inverse of a matrix M notated?

Answer 31

M⁻¹

Question 32

How can the inverse of a 2×2 matrix M with values [(a,b),(c,d)] be calculated?

Answer 32

(1/det(M))*[(d,-b),(-c,a)]

Question 33

What is a singular matrix?

Answer 33

A matrix with no inverse

Question 34

When is a matrix M singular?

Answer 34

When it is not square, or when det(M)=0

Question 35

For two matrices A and B, what is (AB)⁻¹ equal to?

Answer 35

B⁻¹A⁻¹

Question 36

Given the area of an object, A, and a matrix M whose transformation is applied to the object, how can the area of the image be calculated?

Answer 36

A*|det(M)|

Question 37

Given the simultaneous equations ax+by=c and px+qy=r, where a, b, c, p, q and r are constants, how can one solve for x and y using matrices?

Answer 37

Set up the matrix equation [(a,b),(p,q)][(x),(y)]=[(c),(r)]. Premultiplying by [(a,b),(p,q)]⁻¹ leaves [(x),(y)]=[(a,b),(p,q)]⁻¹[(c),(r)]