MST124 BOOK C UNIT 9
- Created by: Scar.Rose.1
- Created on: 09-06-22 12:48
1
- 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 aij + bij
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 aij - bij
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
2
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 kaij
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
3
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 cij denoted the element in the ith row and jth column of AB, then
cij = ai1b1j + ai2b2j + ... + ainbnj
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
4
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
5
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
6
- 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
7
- 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
8
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
Comments
No comments have yet been made