Matrices Year 1 Flashcards
What is the order of the matrix?
Rows x Columns
What is a square matrix?
Same number of row as columns
What is the identity matrix?
1 0
0 1
What is the null matrix?
0 0
0 0
What is the transpose of a matrix?
Writing the columns as rows and vice versa
When can you add and subtract a matrix?
When they have the same order- add and subtract each element
Is matrix multiplication associative?
Yes
Is matrix multiplication commutative?
No
Define object
Original point/shape
Define image
The transformed shape
Define mapping
The transformation
Define i : 1
0
Unit vector in x axis (2D)
What is a unit vector?
A vector with a magnitude of one
Define j : 0
1
Unit vector in y axis (2D)
Define i: 1
0
0
Unit vector in x direction (3D)
Define j: 0
1
0
Unit vector in y direction (3D)
Define k: 0
0
1
Unit vector in z direction (3D)
How would you be able to identify the type of transformation?
e.g what is 2 0
0 2
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
What are the coordinates of the unit square in 2D?
(0,1)
(1,0)
(1,1)
Give the matrix for a enlargement (2D)
a 0
0 a
Where a > 0
Give the matrix of a stretch parallel to x axis (2D)
a 0
0 1
Give the matrix of a stretch parallel to y axis (2D)
1 0
0 a
Give the matrix of a shear parallel to x axis and how would you describe it? (2D)
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)
Give the matrix of a shear parallel to y axis and how would you describe it? (2D)
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)
How do you find the shear factor?
distance between original point and invariant axis
Give the matrix of a reflection in the x axis (2D)
1 0
0 -1
Give the matrix of a reflection in the y axis (2D)
-1 0
0 1
Give the matrix of a reflection in the line y=x (2D)
0 1
1 0
Give the matrix of a reflection in the line y=-x (2D)
0 -1
-1 0
Give the matrix of a reflection in the line x= 0 aka ZY plane (3D)
-1 0 0
0 1 0
0 0 1
Give the unit matrix for a 3D shape
1 0 0
0 1 0
0 0 1
Give the matrix of a reflection in the line y= 0 aka XZ plane (3D)
1 0 0
0 -1 0
0 0 1
Give the matrix of a reflection in the line z= 0 aka XY plane (3D)
1 0 0
0 1 0
0 0 -1
Give the matrix for the anticlockwise rotation about the origin through angle θ (given in formula sheet) (2D)
cosθ -sinθ
sinθ cosθ
What is the order of successive transformations such AB for example?
AB (matrix A X matrix B) means do B first then A
What is an invariant point?
A point that maps to itself under the transformation
a b X x = x
c d y y
How would you find the line of invariant points?
e.g for the matrix
2 -1
1 0
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
What is an invariant line?
Every point on the invariant line maps to either itself or another point on the line
How do you find an invariant line?
e.g find the invariant line on
5 1
2 4
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
How do you find the determinant of a 2x2 matrix?
a b
c d
ad-bc
What does the determinant represent in a transformation?
The area scale factor
What does it mean in the transformation if the determinant is negative?
The orientation of the vertices is reversed
What does |M| mean?
The determinant of matrix m
|MN| =
|M| x |N|
If the determinant is zero what does this mean about the transformation? and how would you find that line?
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
How do you find the inverse of a 2x2 matrix?
a b
c d
1 d -b
——– x
ad-bc -c a
What does it mean if the determinant is zero for its inverse?
It does not have an inverse - it is singular.
(MN) -1 =
M-1 x N-1
How do you find the determinant of a 3x3 matrix?
(a1 x matrix of minor when removing a1 ) - (a2 x matrix of minor when removing a2) + (a3 x matrix of minor when removing a3)
How do you find the inverse of a 3x3 matrix?
1) find matrix of minors
2) put in signs
3) transpose
4) multiply by 1/determinant
How do you solve simultaneous equations using matrices?
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
What are the 3 cases for matrix geometric representation of 2 simultaneous lines?
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)
