Linear System of Simultaneous Equations Warm UP

9 2 :Pr 26 :Pr 1

yxecinctnd

yxecinctst

First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct.

Write a system of two equations and find out how many felonies and misdemeanors occurred.

Sections 4.1 & 4.2

Matrix Properties and Operations

Algebra

Matrix

A

a11 ,, a1n

a21 ,, a2n

am1 ,, amn

Aij

A matrix is any doubly subscripted array of elements arranged in rows and columns enclosed by brackets.

Element

Dimensions of a Matrix

3, 2

Name the Dimensions

Row Matrix

[1 x n] matrix

 

jn aaaaA ,, 2 1

Column Matrix

i

m

a

a

aa

A 2

1

[m x 1] matrix

Square Matrix

B 5 4 73 6 12 1 3

Same number of rows and columns

Matrices of nth order-B is a 3rd order matrix

The Identity

Identity Matrix

I

1 0 0 00 1 0 00 0 1 00 0 0 1

Square matrix with ones on the diagonal and zeros elsewhere.

The Equal

Equal MatricesTwo matrices are considered equal if they have the same dimensions and if each element of one matrix is equal to the corresponding element of the other matrix.

=

Can be used to find values when elements of an equal matrices are algebraic expressions

211039783

211039783

To solve an equation with matrices 1. Write equations from matrix 2. Solve system of equations

Examples

=

=

7

32yx

76

3210 z

3y

xy

x2

23

Linear System of Simultaneous Equations

9 2 :Pr 26 :Pr 1

yxecinctnd

yxecinctst

How can we convert this to a matrix?

Matrix Addition

A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: 

Cij Aij Bij

Note: all three matrices are of the same dimension

Addition

A a11 a12

a21 a22

B b11 b12

b21 b22

C a11 b11 a12 b12

a21 b21 a 22 b22

If

and

then

Matrix Subtraction

C = A - BIs defined by

Cij Aij Bij

Subtraction

A a11 a12

a21 a22

B b11 b12

b21 b22

22222121

12121111

babababa

C

If

and

then

Matrix Addition Example

CBA 10 86 4

4 32 1

6 54 3

CBA 2222

3412

5634

Multiplying Matrices by Scalars

Matrix Operations

Matrix Multiplication

Matrices A and B have these dimensions:

Video

[r x c] and [s x d]

Matrix Multiplication

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s

Matrix Multiplication

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d

A x B = C

A a11 a12

a21 a22

B b11 b12 b13

b21 b22 b23

232213212222122121221121

2312131122121211 21121111

babababababababababababa

C

[2 x 2]

[2 x 3]

[2 x 3]

A x B = C

A 2 31 11 0

and B

1 1 1 1 0 2

[3 x 2] [2 x 3]A and B can be multiplied

1 1 13 1 28 2 5

12*01*1 10*01*1 11*01*132*11*1 10*11*1 21*11*182*31*2 20*31*2 51*31*2

C

[3 x 3]

[3 x 2] [2 x 3]Result is 3 x 3

Inversion

Matrix Inversion

B 1B BB 1 I

Like a reciprocal in scalar math

Like the number one in scalar math

Transpose Matrix

A'

a11 a21 ,, am1

a12 a22 ,, am 2

a1n a2n ,, amn

Rows become columns and columns become rows