Matrices Flashcards
multiplying by the identity matrix
multiplying by 1
multiplying by the null/zero matrix
multiplying by 0
order/dimension of a matrix
the number of rows/columns
you can only add matrices if they have…
the same order
AB != BA
matrix multiplication is not commutative
a(bc) = (ab)c
Associative Property of Multiplication
a(b+c) = ab + ac
Distributive
conformable
mxn order can be multiplied by nxp order giving a matrix of order mxp
a stretch parallel to the x-axis with scale factor k
K 0
0 1
A stretch parallel to the y-axis with scale factor k
1 0
0 K
A enlargement centre of origin with scale factor k
K 0
0 K
A rotation 90 anti-clockwise about the origin
0 -1
1 0
A rotation 90 clockwise about the origin
0 1
-1 0
leading diagonal rule rotation matrices
the elements in the leading diagonal are the exact same
the elements in the opposite diagonal are the same but with opposite signs
A rotation of any angle anticlockwise
cosθ -sinθ
sinθ cosθ
reflection in the x-axis
1 0
0 -1
reflection in the y-axis
-1 0
0 1
reflection in the line y=x
0 1
1 0
reflection in the line y=-x
0 -1
-1 0
Shear
a transformation in which all the point are translated parallel to a particular line by a factor which is proportional to the distance of the point from a shear line
shear parallel to the x-axis
1 K
0 1
shear parallel to the y-axis
1 0
K 1
reflection in the yz plane
-1 0 0
0 1 0
0 0 1
reflection in the xz plane
1 0 0
0 -1 0
0 0 1
rotation of 180 about the z axis
-1 0 0
0 -1 0
0 0 1
Invarient Point
a point which is mapped to itself by the transformation
How to find a line of invarient points
- multiple the matric by x y and make it equal to x y
- multiply out into two equations (top and bottom)
- if (ax + by = x)= (cx +dy = y) then all the points along ax+by = x are invarient points
- else (0,0) is the only invarient point
How to find an invarient line
- multiply the matrix by x y and set it equal to x’ y’
- replace y with mx+c
- replace y’ = mx’ + c
- rearrange and factorise into two parts: the x’s and the c’s
- find the value of m to make the x bracket 0
- sub the value into c’s brack
- IF != 0 then c=0, sub values back into 4. these are the invairent lines
- IF = 0 then c is some contant k, sub these values back into 4, these are invarient linez
determinant
the scale factor of the transformation
determinant of a 2x2
ad-bc
if the determinant is zero the matrix is…
singular
Inverse of a 2x2
1/determinant * d -b
-c a
AA^-1 =
identity matrix
(AB)^-1 =
A^-1 B^-1
determinant of a 3x3
expand a column / row
e.g, multiply each of the elements in a row by its cofactor
inverse of a 3x3 matrix
- find the determinant
- find the cofactor of each element
- replace each element with its cofactor
- reflect in the diagonal to get the adjugate/adjoint matrix (take the transpose)
- divide by the determinant
solving simultaneous equations
- separate out the simultaneous equations into a matrix of the coefficiants multiplied by the unknowns = the constants
- multiply the constants by the inverse of the matrix
what type of solution are there for matrix simultaneous equations?
- no solution (determinant = 0)
- unique solution
- infinitely many solutions
geometric representation of unique solution
planes intersect at a single point
geometric representation of infinitely many solutions
sheaf of planes
geometric representation of no solutions
- three parallel planes
- two planes are parallel and the third is not
- triangular prism
equations for parallel planes
coefficients are the same but constants are different
1 point of intersection (3 simultaneous equations, 3 planes)
detM != 0, non-singular solutions can be found
3 parallel planes
detM = 0, singular, all three equations are coincident
2 parallel planes
detM = 0, singular, two of the equations are coincident
3 equations the same/ multiples (coincident)
3 parallel planes
2 equations same/ multiples (coincident)
2 parallel planes
Sheaf
detM = 0, singular, equations consistent
What is a sheaf
where all three planes cross but instead of a single point of intersection there is a central line down all three points
equations consistent
sheaf (infinitely many solutions)
equations not consistent and not coincident
triangular prism
consistent equations
They can be solved but for infinite solutions, multiples of each other
example of consistent equations
x + y = 1
2x + 2y = 2
example of inconsistent equations
5x - y = 17
5x - y = 15
reflection in y axis function form
f(-x)
reflection in x axis function form
-f(x)