Matrices Flashcards
Non-commutativity meaning
AB != BA
Associativity meaning
(AB)C = A(BC)
Identity meaning
A = AI = IA
Matrices Inverse definition
AA^-1 = I = A^-1 A
Product of inverses, (AB)^-1 =
B^-1 A^-1
Determinant of a 2x2 matrix
ad-bc
Determinant of a 3x3 matrix
a det(efhi) - b det(dfgi) + c det(degh)
If det(M) = 0,
M is singular and has no inverse
Matrix of minors
For every element, draw cross through and find determinant
Matrix of cofactors
Cross +, diamond -, apply to matrix of minors
Transpose of a matrix
Rows and columns swap
2x2 inverse
1/det (d -b
-c a)
3x3 inverse
1/det C^T where C is the matrix of cofactors
Solve matrix equations
Write as (matrix)(x y z) = (answers)
Do inverse to get x y z
Work out what matrix plane equations look like
Check det M, if it isn’t 0 the planes meet at 1 point.
If it is and is consistent, multiple solutions exist like a sheaf
If it is inconsistent and isn’t parallel, it forms a prism
How to work out invariant points
M(x y) = (x y)
How to work out invariant lines, y = mx + c
M(x mx+c) = (x’ y’)
sub back into y = mx + c
Solve
How to work out invariant lines, y = ax
M(x ax) = k(x ax)
What is the area scale factor?
det M
Successive transformations
PQ is Q then P
Rotation x anticlockwise
( cosx -sinx
sinx cosx)
Reflection in plane
3 x 3 identity matrix, turn corresponding column negative
Rotation x anticlockwise about x axis.
1 0 0
0 cosx -sinx
0 sinx cosx
Rotation x anticlockwise about y axis.
cosx 0 sinx
0 1 0
-sinx 0 cosx
Rotation x anticlockwise about z axis
cosx -sinx 0
sinx cosx 0
0 0 1
To undo transformations
P^-1(PQ)Q^-1
Proof by Induction Steps
Base case, n = 1
Assumption, n = k
Inductive case, n = k+1
Conclude true for all n £ N
Divisibility Induction
Use f(k+1) - f(k)
