Matrix algebra Flashcards
What are the dimensions of a matrix?
The number of rows by the number of columns
What property must two matrices have for addition to be possible?
They must have the same dimensions
How can two matrices A and B be added?
Each element in matrix A is simply added to the corresponding value in matrix B
How can a matrix be multiplied by a scalar?
By multiplying each element by the scalar
What property must two matrices A and B have for multiplication to be possible?
The number or columns in A must be equal to the number of rows in B
If matrix A has dimensions a×b and matrix B has dimensions b×c, what are the dimensions of matrix AB?
a×c
How can two matrices A and B be multiplied?
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
What can be said about the commutativity of matrix multiplication?
It is not commutative
What can be said about the associativity of matrix multiplication?
It is associative
What is a transformation?
An operation that moves all points in a plane according to some rule
What is a linear transformation?
A transformation that only involves linear expressions of x and y, and also leaves the origin unmoved
What is a position vector?
The vector from the origin to a point
What is an image?
The point to which a position vector is moved to after a transformation
What can be said about the distributivity of matrix multipliation?
It is distributive
For a linear transformation T:[(x),(y)]→[(ax+by),(cx+dy)], how can T be represented in matrix form?
[(a,b),(c,d)]
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?
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)]
How may the transformation represented by a matrix be found?
By applying it to the vectors [(1),(0)] and [(0),(1)] and observing the ensuing transformations
What matrix represents an anticlockwise rotation of θ around the origin?
[(cos(θ),-sin(θ)),(sin(θ),cos(θ)]
What matrix represents a reflection in the line x=0?
[(-1,0),(0,1)]
What matrix represents a reflection in the line y=0?
[(1,0),(0,-1)]
What matrix represents a reflection in the line y=x?
[(0,1),(1,0)]
What matrix represents a reflection in the line y=-x?
[(0,-1),(-1,0)]
What matrix represents an enlargement of scale factor n around the origin?
[(n,0),(0,n)]
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?
BA
What is the identity matrix?
A square matrix with ones down the main diagonal and zeros elsewhere
How is the identity matrix notated?
I
What property does the identity matrix have?
For any matrix A, AI=IA=A
For a matrix [(a,b),(c,d)], how can the determinant be calculated?
ad-bc
How is the determinant of a matrix M notated?
det(M)
What is the inverse of a matrix M?
The matrix M⁻¹ such that MM⁻¹=M⁻¹M=I
How is the inverse of a matrix M notated?
M⁻¹
How can the inverse of a 2×2 matrix M with values [(a,b),(c,d)] be calculated?
(1/det(M))*[(d,-b),(-c,a)]
What is a singular matrix?
A matrix with no inverse
When is a matrix M singular?
When it is not square, or when det(M)=0
For two matrices A and B, what is (AB)⁻¹ equal to?
B⁻¹A⁻¹
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?
A*|det(M)|
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?
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)]
