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
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
xyx
223
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 9046
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
Practice
Combing Steps
Matrices In the Calculator 2nd x-1 button must enter dimensions before data must enter the matrix before doing calculations
2.5 Determinants of 2 X 2 Matrix
Example Find the value of = 3(9) - 2(5) or 17
3 52 9
Determinants of 3 X 3 MatrixThird-order determinants - Determinants of 3 × 3 matrices are called
Expansion by diagonalsStep 1: begin by writing the first two columns on the right side of the determinant, as shown below
Step 2: draw diagonals from each element of the top row of the determinant Downward to the right. Find the product of the elements on each diagonal.
Determinants of 3 X 3 Matrix
Step 4: To find the value of the determinant, add the products of the first set of diagonals and then subtract the products of the second set of diagonals. The value is:
Step 3: draw diagonals from the elements in the third row of the determinant upward to the right. Find the product of the elements on each diagonal.
Example
Determinants of 3 X 3 Matrixexpansion by minors.
The minor of an element is the determinant formed when the row and column containing that element are deleted.
Example2 -3 -51 2 25 3 -1
2 23 1
1 25 1
1 25 3
.
= 2 -(-3) +(-5)
= 2(-8) + 3(-11) – 5(-7)= -14
Inversion
Matrix Inversion
B 1B BB 1 I
Like a reciprocal in scalar math
Like the number one in scalar math
Inverses
2.5 InversesStep 1 : Find the determinant.Step 2 : Swap the elements of the leading diagonal.Recall: The leading diagonal is from top left to bottom right of the matrix.Step 3: Change the signs of the elements of the other diagonal.Step 4: Divide each element by the determinant.
Example
First find the determinant = 4(2) - 3(2) or 2
or
4 23 2
2 213 42
1 13 22
.
.
Solving Systems with Matrices
Step 1: Write system as matricesStep 2: Find inverse of the coefficient matrix.Step 3: Multiply each side of the matrix
equation by the inverse
Coefficient Matrix
Variable Matrix
Constant Matrix
Example Solve the system of equations by using
matrix equations.3x + 2y = 32x – 4y = 2
3 2 32 4 2
xy
4 2 4 2 31 13 2 2 3 2 3 2162 4
10
ExampleWrite the equations in the form ax + by = c2x – 2y – 3 = 0 2⇒ x – 2y = 38y = 7x + 2 7⇒ x – 8y = –2Step 2: Write the equations in matrix form.
Step 3: Find the inverse of the 2 × 2 matrix.Determinant = (2 × –8) – (–2 × 7) = – 2
Step 4: Multiply both sides of the matrix equations with the inverse
Using the calculator How do you use the calculator to find the
solution to a system of equations? Put both coefficient and answer matrix into
calculator Multiply the inverse of the coefficient matrix and
the answer matrix to get values.
3x + 2y = 3 x + 2y+3z=52x – 4y = 2 3x+2y-2z=-13
5x+3y-z=-11
Modeling Motion with Matrices Vertex Matrix – A matrix used to represent the
coordinates of the vertices of a polygon Transformations -Functions that map points
of a pre-image onto its image Preimage-image before any changes Image-image after changes Isometry-a transformation in which the image
and preimage are congruent figures Translation – a figure moved from one
location to another without cahnging sizze, shape, or orientation
Translation Suppose triangle ABC with vertices A(-3, 1),
B(1, 4), and C(-1, -2) is translated 2 units right and 3 units down.
a. Represent the vertices of the triangle as a matrix.
b. Write the translation matrix. c. Use the translation matrix to find the
vertices of A’B’C’, the translated image of the triangle.
d. Graph triangle ABC and its image.
Translation a. The matrix representing the coordinates of the vertices of triangle
ABC will be a 2 3 matrix.
b. The translation matrix is .
c. Add the two matrices.
d. Graph the points represented
by the resulting matrix.
3 1 1
1 4 2
A B Cx - coordinatey - coordinate
2 2 23 3 3
3 1 1 2 2 2 1 3 11 4 2 3 3 3 2 1 5
A B' C
Example
ΔX'Y'Z' is the result of a translation of ΔXYZ. A table of the translations is shown. Find the coordinates ofY and Z'.Solve the Test Item• Write a matrix equation. Let (a, b) represent the coordinates of Y and let (c, d) represent thecoordinates of Z'.• Since these two matrices are equal, corresponding elements are equal.Solve an equation for x. Solve an equation for y.–3 + x = 4 2 + y = 7x = 7 y = 5• Use the values for x and y to find the values for Y(a, b) and Z' (c, d).a = –4 b = –5 9 = c 4 = d
ΔXYZ ΔX'Y'Z'X(–3, 2) X'(4, 7)Y Y'(3, 0)Z(2, –1) Z'
Dilation When a geometric figure is enlarged or
reduced ALL linear measures of the image change
in the same ration
ExampleQuadrilateral DEFG has vertices D(1, 2), E(4, 1), F(3, –1), and G(0, 0). Dilate quadrilateral DEFG so that its perimeter is two and one–half times the original perimeter. What are the coordinates of the vertices of quadrilateral D'E'F'G'?If the perimeter of a figure is two and one–half times the original perimeter, then the lengths of the sides of the figure will be two and one–half times the measure of the original lengths.Multiply the vertex matrix by the scalar 2.5.
The coordinates of the vertices of quadrilateral D'E'F'G' are D'(2.5, 5), and E'(10, 2.5), F'(7.5, –2.5),and G'(0, 0).
2.4 Modeling Motion with Matrices
Reflection Matrices X-Axis1 00 -1 Y-Axis
Line y=x0 11 0
Reflection Example Find the coordinates of the vertices of the
image of quadrilateral ABCD with A(–2, 1), B(–1, 4), C(3, 2), and D(4, –2) after a reflection across the line y = x.
Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the line y = x.
The coordinates of the vertices of A'B'C'D' are A'(1, –2), B'(4, –1), C'(2, 3), and D'(–2, 4). Notice that the preimage and image are congruent. Both figures have the same size and shape.
2.4 Modeling Motion with Matrices Rotation Matrices 90 degrees
180 degrees
270 degrees
Rotation Example Find the coordinates of the vertices of the image
of quadrilateral MNOP with M(2, 2), N(2, 5), O(3, 4), and P(4, 1) after it is rotated 270° counterclockwise about the origin.
Write the ordered pairs in a vertex matrix. Then multiply the vertex matrix by the rotation matrix.
The coordinates of the vertices of quadrilateral M'N'O'P' are M'(2, –2), N'(5, –2), O'(4, –3), and P'(1, –4). The image is congruent to the preimage