Linear System of Simultaneous Equations Warm UP

9 2 :Pr 2

6 :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

a

a

A 2

1

[m x 1] matrix

Square Matrix

B

5 4 7

3 6 1

2 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 0

0 1 0 0

0 0 1 0

0 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

211

039

783

211

039

783

To solve an equation with matrices

1. Write equations from matrix 2. Solve system of equations

Examples

=

=

7

32

y

x

76

3210 z

3

y

xy

x

2

23

Linear System of Simultaneous Equations

9 2 :Pr 2

6 :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

baba

babaC

If

and

then

Matrix Addition Example

CBA 90

46

4 3

2 1

6 5

4 3

CBA 22

22

34

12

56

34

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

babababababa

babababababaC

[2 x 2]

[2 x 3]

[2 x 3]

A x B = C

A

2 3

1 1

1 0

and B

1 1 1

1 0 2

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

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*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 5

2 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.  

Example

2 -3 -5

1 2 2

5 3 -1

2 2

3 11 2

5 11 2

5 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 2

3 2

2 21

3 42

1 1

32

2

.

.

Solving Systems with Matrices

Step 1: Write system as matrices

Step 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 = 3

2x – 4y = 23 2 3

2 4 2

x

y

4 2 4 2 31 13 2 2 3 2 3 216

2 4

1

0

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=5

2x – 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 C

x - coordinate

y - coordinate

2 2 2

3 3 3

3 1 1 2 2 2 1 3 1

1 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-Axis

1 0

0 -1 Y-Axis

Line y=x

0 1

1 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