linear system of simultaneous equations warm up
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Linear System of Simultaneous Equations Warm UP. 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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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Sections 4.1 & 4.2
Matrix Properties and Operations
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Algebra
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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
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Name the Dimensions
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Row Matrix
[1 x n] matrix
jn aaaaA ,, 2 1
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Column Matrix
i
m
a
a
aa
A 2
1
[m x 1] matrix
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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
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The Identity
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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.
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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
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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
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Linear System of Simultaneous Equations
9 2 :Pr 26 :Pr 1
yxecinctnd
yxecinctst
How can we convert this to a matrix?
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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
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Addition
A a11 a12
a21 a22
B b11 b12
b21 b22
C a11 b11 a12 b12
a21 b21 a 22 b22
If
and
then
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Matrix Subtraction
C = A - BIs defined by
Cij Aij Bij
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Subtraction
A a11 a12
a21 a22
B b11 b12
b21 b22
22222121
12121111
babababa
C
If
and
then
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Matrix Addition Example
CBA 10 86 4
4 32 1
6 54 3
CBA 2222
3412
5634
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Multiplying Matrices by Scalars
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Matrix Operations
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Matrix Multiplication
Matrices A and B have these dimensions:
Video
[r x c] and [s x d]
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Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s
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Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d
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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]
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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
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Inversion
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Matrix Inversion
B 1B BB 1 I
Like a reciprocal in scalar math
Like the number one in scalar math
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Transpose Matrix
A'
a11 a21 ,, am1
a12 a22 ,, am 2
a1n a2n ,, amn
Rows become columns and columns become rows