# 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 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

## 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 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

## 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 c

_{ij}denoted the element in the ith row and jth column of AB, then

c

_{ij}= a

_{i1}b

_{1j}+ a

_{i2}b

_{2j}+ ... + a

_{in}b

_{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

## 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 A^{2} 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 A^{n} = 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

^{-1}A = 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 X_{11} = a, X_{12} = b, X_{21} = c and X_{22} = 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

^{-1}b

## 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

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