MST124 BOOK C UNIT 9

- a matrix with m rows and n columns is described as an m x n matrix 
- rows x columns 
- rows are i, columns are j 

Matrix addition
if A and B are m x n matrices, then A+B is the m x n matric whose element in row i and column j is a_ij + b_ij
Addition of two matrices of different sizes is not defined 

Matrix subtraction
If A and B are m x n matrices, then A - B is the m x n matrix whose element in row i and column j is a_ij - b_ij 
Subtraction of two matrices of different sizes is not defined 

- a matrix each of whose elements is 0 is called a 0 matrix, denoted by 0 
- if A is any matrix, then the matrix formed by changing each element of A to its negative is called the negative of A, denoted by -A 

Scalar Multiplication
If A is an m x n matrix and k is any real number, then kA is the matrix whose element in row i and column j is ka_ij

Properties of matrix addition and scalar multiplication
A + B = B + A
(A + B) + C = A + (B + C)
A + 0 = A
A + (-A) = 0
m(A + B) = mA + mB
(m + n)A = mA + nA
m(nA) = (mn)A
1A = A 

Matrix Multiplication
Let A and B be matrices. Then the product matrix AB can be formed only if the number of columns of A is equal to the number of rows of B.
If A has size m x n and B has size n x p, then the product AB has size m x p
The element in row i and column j of the product matrix AB is obtained by multiplying each element in the ith row of A by the corresponding element in the jth column of B and adding the results
In element notation, if c_ij denoted the element in the ith row and jth column of AB, then
c_ij = a_i1b_1j + a_i2b_2j + ... + a_inb_nj 

Some properties of matrix multiplication
The following properties hold for all matrices A, B and C for which the products and sums mentioned are defined
(AB)+C = A(BC)
k(AB) = (kA)B = A(kB), for any scalar k
A(B+C) = AB + AC 
(A+B)C = AC + BC 

Matrix multiplication is not commutative
- there are matrices A, B such that the product AB exists but the product BA does not
- even if both products are defined, it can happen that AB does not equal BA 

- the square A² of the matrix A is defined to be the matrix product AA
- in general, for any square matrix A, the nth power of A is Aⁿ = AA...A (number of A equal n)

- dots in a network diagram are called nodes
- lines are called arcs 

- to convert a network to a matrix, the inputs are the columns and the outputs are the rows 

- networked can be combined by making the outputs from one network become the inputs to another network
- these types of networks can be simplified by drawing a network that uses only the original inputs and the final outputs 
- the labels for the arcs are the value for each individual arc in the original network multiplied by each other, and then

INPUT = COLUMNS
OUTPUT = ROWS

- an identity matrix is a matrix that behaves like the number 1
- if a matrix A is multiplied by an identity matrix of an appropriate size, then the result is again A 
- identity matrix denoted by I 
- if A is any matrix such that the product AI exists, then AI = A
- if A is any matrix such that the product IA exists, then IA = A
- any matrix I with these two properties is called an identity matrix 
- identity matrices must be square
- in order for the product AI to be defined, if A is an m x n matrix, then I must have n rows 
- in order for AI to have size m x n, which it must if equal to A, I must have n columns 
- I must have size n x n, relative to m x n of A 

- for every natural number n, the n x n matrix that has ones down the leading diagonal (starting in top left corner), and zeros everywhere else is an identity matrix 
- for every natural number n, this matrix is the only n x n identity matrix 

- let A be a square matrix
- if there is another matrix B of the same size with the property that AB = I and BA = I where I is an identity matrix
- A is invertible and B is an inverse of A 

- if a matrix A is invertible, then it has exactly one inverse
Inverse of a matrix
If A is an invertible matrix, then AA^-1 = I and A^-1A = 1
where I is the identity matrix of the same size as A

Inverse of a 2 x 2 matrix
The inverse of the 2 x 2 (X) with elements X₁₁ = a, X₁₂ = b, X₂₁ = c and X₂₂ = d, is given by 1/(ad-bc) * the 2 x 2 matrix, where the elements on the leading diagonal are swapped and the other two elements are multiplied by -1 

Determinant of a 2 x 2 matrix
Using the previously described 2 x 2 matrix, the number ad - bc is called the determinant of A, also written as det A. If detA is equal to 0, then A is not invertible, otherwise A is invertible 

- to work out the determinant of larger matrices, use a computer algebra program 

Matrix form of simultaneous linear equations
The simultaneous linear equations ax + by = e and cx + dy = f can be written as a matrix equation
(a b) (x) = (e)
(c d) (y)     (f)
- the 2 x 2 matrix on the LHS is called the coefficient matrix 

Strategy
To solve a pair of simultaneous linear equations in two unknowns using matrices
Write the simultaneous linear equations in matrix form Ax = b, where A is the coefficient matrix, c is the corresponding vector of unknowns, and b is the vector whose components are the corresponding right-hand sides of the equations.
If the matrix A is invertible, then the solution is given by
x = A^-1b 

For a pair of simultaneous linear equations in two unknowns:
- if the determinant of the coefficient matrix is non-zero, then this matrix is invertible and the equations have a unique solution
- if the determinant of the coefficient matrix is zero, then this matrix is not invertible and the equations have no solutions or infinitely many solutions