Linear System of Simultaneous Equations Warm UP
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Transcript of Linear System of Simultaneous Equations Warm UP
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
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.
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