Converting to minutes and seconds

When converting from minutes to seconds you: 1- take the numbers before the decimal, which is the degrees part of he answer 2- multiply the numbers after the decimal by 60 3- take the numbers before the decimal after you multiply, which gives you the minutes part of your answer 4- multiply the numbers after the decimal from the previous number by 60 5- take th numbers before the decimal, if there is one, which is the seconds part of your answer Example 1- Convert 11.31 to degrees, secondnds, and minutes. 11 degrees 31*60=678 minutes 6*60=360 seconds 11 degrees, 678 minutes, 360 seconds Example 2- Convert 6.12 to degrees, secondnds, and minutes. 6 degrees 12*60=720 There is no decimal, so there is no seconds in this answer. 6 degrees, 720 minutes Example 3- Convert 34.19 to degrees, secondnds, and minutes. 34 degrees 19*60=1140 minutes There is no decimal, so there is no seconds in this answer. 34 degrees, 1140 minutes

Degrees, Minutes, and Seconds!

Merry Christmas!!

To convert to degrees minutes and seconds take the decimal and multiply it by 60, this number will be the minutes, than take that decimal point and multiply it by 60, this number will be the seconds.

___________________________________________________________________

___________________________________________________________________

Ex.1: Convert 12.22 degrees to degrees mins and secs.

.22*60=13.2
.2*60=12
12 degrees 13 minutes 12 seconds

Ex.2: Convert 125.325 to degrees, minutes, and seconds.

.325*60= 19.5 minutes.
.5*60= 30 seconds.
125 degrees 19 minutes 30 seconds

Ex.3:Convert 199.027 to degrees, minutes, and seconds.
.027*60= 1.62 minutes.
.62*60= 37.2 seconds.
199 degrees 1 minute 37 seconds

degrees and radians

Convert 12.22 degrees to degrees mins and secs.
.22*60=13.2
.2*60=12
12 degrees 13 minutes 12 seconds

 Convert 20 degrees 20 minutes 6 seconds to degrees.
20+20/60+6/3600=20.335 degrees

Convert 196 degrees to radians.
196 degrees* pie/180=49/45pie

To convert to minutes you take what is behind the decimal and multiply by 60.  To convert to seconds you take what is behind the decimal in the minutes and multiply it by 60.  In the first problem I took the .22 and multiplied it by 60 and got 13.2.  Now you take the .2 and multiply that by 60 and you get 12.  So now you put it together and your answer is 12 degrees 13 minutes and 12 seconds.  To convert from minutes and seconds to degrees you have to divide the minutes by 60 and the seconds by 3600 and you get 20.335 degrees.  To convert 196 degrees to radians, you have to multiply that number by pie/180.  In the third problem, I multiplied 196 degrees times pie/180 and I got 49/45 pie as my answer.

matrices

When
multiplying matrices (a)(b)x(b)(c) the inner numbers have to from both matrices, and the
resulting matrix will be a(c), the outer numbers left from the matrices.


Example 1:
Find
each matrix product.
[-6 [-1 12 = Not
Defined
0] 0 -4]
2x1 2x2
The inner numbers of both matrices are not the same numbers so there for this matrix cannot be multiplied so it is undefined.

Example 2:

Find each matrix product.
[-2
3 [0 3 = [-18
9
4 2] -6 5] -12 32]
2x2 2x2
-2(0)+3(-6)=-18
-2(3)+3(5)=9
4(0)+2(-6)=-12
4(3)+4(5)=32

Exapmle
3:
Find
each matrix product, if it is defined.
[8 -10 [-2 = Not
Defined
0 3 -9
-6 4] 1]
3x2 3x1
The inner numbers of both matrices are not the same numbers so there for this matrix cannot be multipled so it is undefined.

Hope everyone had a good christmas(:

Parabolas!

MERRRYYY CHRISTMASS EVERYONEE!
--Hope everyone had a safe/blessed day! (:

-Only have x^2 or y^2, not both
-Axis of symmetry: x=-b/2a
-Vertex: (-b/2a, f(-b/2a))
-Focus:
1. Find p using the formula: 1/4p=coeff of leading term
2. Vertex coordinate +p=your focus if it is x^2, you add the y coordinate of the vertex and vise versa.
3. Put in point form

--Standard Form: y+h=a(x+k)
Vertex(k,-h)
a-leading coeff.

Directrix:
1. Find p (see above)
2. Vertex coordinate -p=directrix
3. Write as y=number

Example 1:
For each parabola give the coordinates of its vertex and focus and the equation of its directrix.
y=1/8x^2
x=-0/2(1/8)
x=0
vertex: 1/8(0)^2
(0,0)
1/4p=1/8
p=2
0+2=2
(0,2)
0-2=-2
y=-2

Example 2:
For each parabola give the coordinates of its vertex and focus and the equation of its directrix.
x=1/8y^2
y=0/2(1/8)
y=0
vertex: 1/8(0)^2
(0,0)
1/4p=1/8
p=2
0+2=2
(2,0)
0-2=-2
x=-2

Example 3:
For each parabola give the coordinates of its vertex and focus and the equation of its directrix.
y=-1/12x^2
x=0/2(-1/2)
x=0
Vertex: -1/12(0)^2
(0,0)
1/4p=-1/12
-4p=12
p=-3
0+-3=-3
(0,-3)
0+3=3
y=3

Review of all the Formulas :)

MERRY CHRISTMAS everyone, hope everyone got what they wanted from Santa :)

Chapter 6
-Circles (x - h)^2 + (y - k)^2 = r^2
-Ellipes x^2 over a^2 + y^2 over b^2 = 1
-Hyperbolas x^2 over a^2 - y^2 over b^2 = 1
-To find the shape of the graph b^2 - 4ac

Chapter 7
-Convert degrees to radians: degrees = pi over 180
-Convert radians to degrees: rads x 180 over pi = degrees
-k= 1/2r^2(theta)
-k= 1/2rs
-s= r(theta)
-Unit Circle= extremely important!

Chapter 8
-m= tan(alpha)
-csc(theta)=1/sin(theta)
-sec(theta)=1/cos(theta)
-cot(theta)=1/tan(theta)
-sin^2(theta) + cos^2(theta) = 1
-1 + tan^2(theta) = sec^2(theta)
-1 + cot^2(theta) = csc^2(theta)

Chapter 9
-SOHCAHTOA
-sin(theta)= opp/hyp
-cos(theta)= adj/hyp
-tan(theta)= opp/adj
-sin A/a = sinB/b = sinC/c
-Right triangle: 1/2bh
-Non right triangle: 1/2(adj leg)(adj leg)sin(angle below)
-Law of Cosines: leg^2= adj leg^2 + other adj leg^2 - 2(adj leg)(other leg)cos(angle below)

Chapter 10
-cos(alpha plus or minus beta) = cos(alpha)cos(beta) minus or plus sin(alpha)sin(beta)
-sin(alpah plus or minus beta)= sin(alpha)cos(beta) plus or minus cos(alpha)sin(beta)
-tan(alpha+beta) = tan(alpha + beta)/1-tan(alpha)tan(beta)
-tan(alpha-beta) = tan(alpha-beta)/1+tan(alpha)tan(beta)
-sin2(alpha)= 2sin(alpha)cos(alpha)
-cos2(alpha)= cos^2(alpha)-sin^2(alpha)
-cos2(alpha)= 1-2sin^2(alpha)
-cos2(alpha)= 2cos^2(alpha)-1
-tan2(alpha)= 2tan(alpha)/1-tan^2(alpha)
-sin(alpha)/2= plus or minus square root of 1-cos(alpha)/2
-cos(alpha)/2= plus or minus square root of 1+cos(alpha)/2
-tan(alpha)/2= plus or minus square root of 1-cos(alpha)/1+cos(alpha)
-tan(alpha)/2= sin(alpha)/1+cos(alpha)
-tan(alpha)/2= 1-cos(alpha)/sin(alpha)

Chapter 11
-Convert polar to rectangular: x=rcos(theta) y=rsin(theta)
-Convert rectangular to polar: r= square root of x^2 + y^2 theta=tan-1(y/x)

Keep having a good holidays everyone :) -Danielleeeee

Multiplying Matricies

Multipling matricies is done very simpily. You can mutiply them by a single number or another matrix. There are two sets of rules to follow, one for each type of multiplication.

1.To multiply a matrix by another single number is very simple. You just take the number given and multiply each number in the matrix by that number. You must watch out for negartives for they can be tricky if not played close attention too.

2.To multiply another matrix you must make sure they match. The two inside number of the demensions side by have to be the same EX A*B B*C. The two outer numbers will be the demensions of your new matrix. EX A*B B*C (New demensions A*C). Then you mutliply each row by each column and add all sums. Then put into new matrix.

Ex1. 4*(9 3) = (32 12)
(0 -1) (0 -4)


Ex2. (2 0) * (1 5)
(3 4) (2 -1)
2(1)+0(2)=2
2(5)+0(-1)=10
3(1)+4(2)=15
3(5)+4(-1)=11
(2 10)
(15 11)

circles

find the center and radius:
x^2+y^2-6x+4y-12=0
x^2-6x+__+y^2+4y+__=12
x^2-6x+9+y^2+4y+4=12+9+4
(x-3)^2+(y+2)^2=25
C: (3,-2)  R=5

To start off this problem you have it in standard form.  Standard form for a circle is (k-h)^2+(y-k)^2=r^2. Then, for this problem they are asking you for the center and radius.  To find the radius you can half the diameter found by using 2 points on the outside through the center.  The equation we have is in standard form, so now you have to half the middle term, then square it and add that number to both sides.  So we took 6 and divided it by 2, then squared that number and added it to both sides of the equation.  You do the same thing with the 4 and add it to both sides.  Now your equation looks like (x-3)^2+(y+2)^2=25.  To find the center you take the opposite signs of the numbers and parentheses and make it into a point.  So your center is (3,-2).  To find the radius you just take the 25 and square root it to get 5.

Determinates-Danielle

Number 1

[0 3]

[2 6]

0-6

-6

Number 2

[11 1]

[10 0]

0-10

-10

Number 3

[12 12]

[3 4]

48-36

12

Number 4

[0 0 0]

[1 1 1]

[2 2 2]

0

Merry Christmas ;)

Determinants-kaylaaaa

To find the determinant of a 2 by 2 matrix, you multiply A and D and subtract it from B multiplied by C.

[a b]

[c d]

Number 1

[2 2]

[1 6]

2(6)-2(1)

12-2

10

Number 2

[2 4]

[-6 8]

2(8)-(-6)(4)

16+24

40

Number 3

[2 1 0]

[0 2 3]

[1 0 5]

(2(2)(5)+1(3)(1))

20+3

23

Determinants!

~Notation- | |similar to absolute value

~If the determinant =0 then it is called a singular matrix because it doesn't have an inverse.

~For a 2X2 matrix you use ad-bc

~For a 3X3 matrix - recopy the first 2 columns, then multiply the diagonals add the right, subtract the left

~For a 4X4 matrix it may be easier to just use your calculator.

__________________________________________________________________

Ex.1:

find the determinant of:

|2 6|

|1 9|

2(9)-6(1)=12

Ex.2:

find the determinant of:

|5 6 4|

|0 4 7|=-162

|3 8 2|

Ex.3:

find the determinant of:

|1 9 6 4|

|2 5 4 3|=-123

|7 0 8 6|

|5 2 9 8|

Ex.4:

find the determinant of:

|12 3|=174

|2 15|

Ex.5:

find the determinant of:

|2 13 8|

|11 6 3|=-963

|7 5 10|

Ex.6:

find the determinant of:

|0 1 2 3|

|2 0 1 4|=25

|6 2 3 1|

|1 0 0 3|

Cramer's Rule!

Well, here goes the first blog of the HOLIDAYSSSS! :)

x=Dx/D y=Dy/D z=Dz/D

Where D is the determinant of the coefficient matrix.

Used to solve systems of equations using matrices and determinants.

Example 1:
Solve each system of equations by using Cramer's Rule.
5x-4y=1
3x+2y=5
D=5 -4 =22 Dx=1 -4 =22 Dy=5 1 =22
3 2 5 2 3 5
x=22/22=1 y=22/22=1
(1,1)

Example 2:
Solve each system of equations by using Cramer's Rule.
5x-2y=11
x+3y=9
D=5 -2 =17 Dx=11 -2 =51 Dy=5 11 =34
1 3 9 3 1 9
x=51/17=3 y=34/17=2
(3,2)

Example 3:
Solve each system of equations by using Cramer's Rule.
3x+2y=-1
2x-y=4
D=3 2 =-7 Dx=-1 2 =-7 Dy=3 -1 =14
2 -1 4 -1 2 4
x=-7/-7=1 y=14/-7=-2
(1,-2)

Example 4:
Solve each system of equations by using Cramer's Rule.
7x+y=7
-x+2y=14
D=7 1 =15 Dx=7 1 =0 Dy=7 7 =105
-1 2 14 2 -1 14
x=0/15=0 y=105/15=7
(0,7)

Example 5:
Solve each system of equations using Cramer's Rule.
x-2y+3z=2
2x-3y+z=1
3x-y+2z=9
D= 1 -2 3 =18 Dx= 2 -2 3 =54 Dy= 1 2 3 =36 Dz= 1 -2 2 =18
2 -3 1 1 -3 1 2 1 1 2 -3 1
3 -1 2 9 -1 2 3 9 2 3 -1 9
x=54/18=3 y=36/15=2 z=18/18=1
(3,2,1)

&&& of course my matrices WOULD mess up once again.. I still don't know how to fix themmmm :(

matrix multiplication

Multiplying matrices with the same dimensions

In order to multiply a matrix the inner numbers have to be the same.

-For example a 2x3 and 3x3 would be possible but a 3x4 and 2x4 wouldn’t be possible.

The outer numbers would give you the answer’s dimensions.

-For example a 2x3 and 3x3 would give you a 2x3 matrix.

Solve by multiplication.

1. [ 4 3] [5]

[-1 -2] [1]

The dimensions of this matrix are 2x2 and 2x1. This means this matrix is possible to multiply. First you multiply the first row by the first column. So, 4(5)+3(1)=23. Then multiply the second row by the by the column. So, -1(5)+-2(1)=-7. So the answer would be [23]

[-7]

2. [-6] [-1 12]

[ 2] [ 0 -4]

This matrix cannot be multiplied because the dimensions are 2x1 and 2x2. The inner numbers are not the same which means they cannot be multiplied.

Finding Theta

Solving for theta is differevt from solving for a variable in algebra. The same concepts are used in some ways. There are steps to solving for theta
1.Isolate the trig function
2.Take the inverse of the trig function. This means find the angle.
ex: cos-1 sin-1
arc cos arc cos
*an increase finds an angle
3.Use the trig chart or calculator to find the answer (only use the value)
4.Use the 4 quadrants to find the right angle weather its a + number or - number with any trig function given.
Important
*there are atleast 2 answers for each inverse or equation
*you must know steps to move between quadrants
*sometimes you will have to use unit circle

Steps to move between quadrants:
Q1 - Q2 make # -and add 180
Q1- Q3 add 180
Q1 - Q4 mkae # - and add 360

Ex1
Cos -1 (1/2)
The value of cos (1/2) on trig chart is 60
Since the (1/2) is positive the number stays positive
Now you must find the other positive answer. Cos is also positive in quadrant 4 so you must move 60 to quadrant 4.
-60 + 360 = 300
Your two answers are (60, 300)

Determinants

Determinates

When you see a || around matrix, it is telling you to find the determinant. 

Finding a determinant is the same thing as finding the absolute value of a matrix.

Singular Matrix: when the determinant is equal to zero and it does not have an inverse.  

2x2 Matrix Formula: ad-bc 

3x3 Matrix Formula: Recopy the first 2 columns then multiply all the diagonals and add the ones on the right and subtract the ones on the left. 

Example 1:

(4 1)

(2 3) 

=4(3)-1(2)

=12-2

=10

Example 2:
(5 1)
(3 2)
=5(2)-1(3)
=10-3
=7

MATRICIES

WAZZAM WAZZAM WAZZAM it's ya boy reporting live from his house, bout to smash this matrices blog up because it was just that easy of a section

A^t means transpose ----> switch rows and columns.

To add matrices they must be the same dimension (row x column)

To complete scaler multiplication you multiply every entry by the number

Example 1:

[ 3 1 5 ]

4 0 -2

a. What is the dimension?

2 x 3 matrix

b. Find M^t (transpose)

[3 4 ]

1 0

5 -2

c. Find 2m (multiply)

[ 6 2 10 ]

8 0 -4

1) To find the dimension of a matrix, you have to count how many columns it has (2) and how many rows it has (3)

2) To find M^t (transpose) you have to switch the rows & columns, which will make the example matrix a 3x2 when transposed.

3) To find 2M, you have to multiply every single number in the matrix by two.

*Calculator*

1) press second matrix

2) go over to edit

3) type in your matrix

Muliplying Matrixs.

Ohh Brob thanks for alllll the wonderful study guides :)

So sometime this week…maybe last week we learned how to multiply matrices:

When multiplying matrices (a)(b)x(b)(c) the inner numbers have to match, and the resulting matrix will be a(c)


Example 1:
Find each matrix product, if it is defined.
[-6 [-1 12 = Not Defined
0] 0 -4]
2x1 2x2
*inner numbers do not match, so it is not defined

Example 2;

Find each matrix product, if it is defined.
[-2 3 [0 3 = [-18 9
4 2] -6 5] -12 32]
2x2 2x2
-2(0)+3(-6)=-18
-2(3)+3(5)=9
4(0)+2(-6)=-12
4(3)+4(5)=32

Exapmle 3:
Find each matrix product, if it is defined.
[8 -10 [-2 = Not Defined
0 3 -9
-6 4] 1]
3x2 3x1
*inner number do not match, so it is not defined

--Bloggers poo and messes up the matrices. –Danielle :)

Matrices!! My blog is better than yours.. :)

This is probably one of the easiest sections we have learned all year in Advanced Math so be ready for an AWESOME blog ;)

First we will learn to multiply matrices:

(a)(b)x(b)(c)
The inner numbers have to match, or you can simply not do it. If you even attempt your an idiot.
The answer to this matrices will be a(c).

Example 1:
Find each matrix product, if it is defined.
[-3 [-4 1 =Not Defined
1] 1 -6]
2x1 2x2
The inner numbers don't match so you cant do it! Hahaha tricked ya! :-P

Example 2:
Find each matrix product, if it is defined.
[0 1 [5 2 = [4 9
1 3] 4 1] 17 5] This is your answer! It is easy I know :)
2x2 2x2

Example 3:
Find each matrix product, if it is defined.
[1 -1 [-3 =Not Defined
2 6 -7
-5 4] 0]
3x2 3x1

Once again the inner numbers don't match so this one you can't do, so don't try.

Determinants

~Notation- | | kind of like absolute value

~If the determinant =0 then it is called a singular matrix because it doesn't have an inverse.

~For a 2X2 matrix the formula is ad-bc

~For a 3X3 matrix - recopy the first 2 columns, then multiply the diagonals add the right, subtract the left

~For a 4X4 matrix it may be easier to just use your calculator.
__________________________________________________________________

Ex.1:
find the determinant of:
|4 6|
|3 7|
4(7)-6(3)=10

Ex.2:
find the determinant of:
|5 4 8|
|0 3 7|=-36
|1 2 2|

Ex.3:
find the determinant of:
|2 4 7 9|
|6 5 1 0|=638
|1 9 6 8|
|2 8 5 9|

Ex.4:
find the determinant of:
|7 4|=-12
|3 0|

Ex.5:
find the determinant of:
|7 5 1|
|4 8 3|=118
|3 7 6|

Ex.6:
find the determinant of:
|8 9 6 5|
|5 8 4 3|=-112
|0 1 6 7|
| 9 3 6 5|

Determinants

~Notation- | | kind of like absolute value

~If the determinant =0 then it is called a singular matrix because it doesn't have an inverse.

~For a 2X2 matrix the formula is ad-bc

~For a 3X3 matrix - recopy the first 2 columns, then multiply the diagonals add the right, subtract the left

~For a 4X4 matrix it may be easier to just use your calculator.
__________________________________________________________________

Ex.1:
find the determinant of:
|4 6|
|3 7|
4(7)-6(3)=10

Ex.2:
find the determinant of:
|5 4 8|
|0 3 7|=-36
|1 2 2|

Ex.3:
find the determinant of:
|2 4 7 9|
|6 5 1 0|=638
|1 9 6 8|
|2 8 5 9|

Ex.4:
find the determinant of:
|7 4|=-12
|3 0|

Ex.5:
find the determinant of:
|7 5 1|
|4 8 3|=118
|3 7 6|

Ex.6:
find the determinant of:
|8 9 6 5|
|5 8 4 3|=-112
|0 1 6 7|
| 9 3 6 5|

To find the final answer use the following:

x=Dx/D y=Dy/D z=Dz/D

These are used to solve systems of equations using matrices and determinants

Where D is the determinant of the coefficient matrix

Solve the system of equations.

1. 5x-4y=1

3x+2y=5

First you find the determinant which you copy the x’s in a column and the y’s in a column. This gives you (5 -4 and the answer is 22.

3 2)

You then replace the x’s with the answers 1 and 5. So Dx= (1 -4 and the answer is 22.

5 2)

Then replace the y’s with the answers 1 and 5. So Dy= (5 1 and the answer is 22.

3 5)

You then use the formulas from above. X=22/22 which is 1, so x equals 1. Y=22/22 which is 1, so y equals 1.

2. 5x-2y=11

x+3y=9

First you find the determinant which you copy the x’s in a column and the y’s in a column. This gives you (5 -2 and the answer is 17.

1 3)

You then replace the x’s with the answers 11 and 9. So Dx= (11 -2 and the answer is 51

9 3)

Then replace the y’s with the answers 11 and 9. So Dy= (5 11 and the answer is 34.

1 9)

You then use the formulas from above. X=51/17 which is 3, so x equals 3. Y=34/17 which is 2, so y equals 2.

Determinants

When you see || around a matrix, that means find the determinant

(same things as absolute value)

For a 2x2 matrix:

|a b|

|c d|

To find determinant, ad - bc

For a 3x3, copy the 1st 2 columns of the matrix right next to it and

cross multiply each diagonal, then add and subtract results.

For a 4x4, just use a calculator because it's really complicated.

___________________________________________

Examples:

1. |3 2|

|5 1|

3-10= -7

2. |1 2 4| 1 2

|5 3 1| 5 3

|0 7 2| 0 7

1(3)2 + 2(1)0 + 4(5)7 - 2(5)2 - 1(7)1 - 4(3)0 = 119

Determinants!

~Notation- | |similar to absolute value

~If the determinant =0 then it is called a singular matrix because it doesn't have an inverse.

~For a 2X2 matrix you use ad-bc

~For a 3X3 matrix - recopy the first 2 columns, then multiply the diagonals add the right, subtract the left

~For a 4X4 matrix it may be easier to just use your calculator.

__________________________________________________________________

Ex.1:

find the determinant of:

|2 6|

|1 9|

2(9)-6(1)=12

Ex.2:

find the determinant of:

|5 6 4|

|0 4 7|=-162

|3 8 2|

Ex.3:

find the determinant of:

|1 9 6 4|

|2 5 4 3|=-123

|7 0 8 6|

|5 2 9 8|

Ex.4:

find the determinant of:

|12 3|=174

|2 15|

Ex.5:

find the determinant of:

|2 13 8|

|11 6 3|=-963

|7 5 10|

Ex.6:

find the determinant of:

|0 1 2 3|

|2 0 1 4|=25

|6 2 3 1|

|1 0 0 3|

Review of chapter 8-4

Chapter 8-4 was the hardest section for me so i decided to review it.

Relationships among functions.
To find relationships among functions you use identities. Identities is another word for a bunch of formulas that show how trig functions and can be used with other trig functions and how they relate. Usisng these formulas you follow the steps and do some algebra to come up with a more simplified version of the problem. You will sometimes be asked to simplify wear you just simplify the equation down to the smalles you can get it or will be asked to prove which means you only have to show the steps of how to get the answer they've already given you. These are the steps that you should follow.
1.Algebra
2.Identities-try pythagorean first. If it doesnt work try to change all to sin and cos
3.Algebra
4.Identities and so on repeat until all the way simplified

Identities:
Pythagorean
sin^2@+cos^22=1 1+tan^2@=sec^2@

1+cot^2@=csc^2@

Reciprical
csc@=1/sin@ secf=1/cos@

cot@=1/tan@

Cofunction
sin@=cos(90-@) cos@sin(90-@)
tan@=cot(90-@) cot@=tan(90-@)
sec@=csc(90-@) csc@=sec(90-@)

Ex1
sec^2@-1
This goes with pythagorean formula. If rearraged formulas states that 1+tan^2@=sec^2@
There fore this equation is simplified to tan^2@

Determinates

1   -3   4 line 1   -3
 0   1   1        0   1
5   -2   3       5    -2

3+-15-2+-20=-32

This is a 3 by 3 matrix.  To know how to identify a matrix, all you do is look how many rows they have in the matrix, and then look how many columns there is in the matrix. Then, you just write it down like this:  3*3.  Now to find the determinate for a 3 by 3 matrix all you write down the first two columns in the matrix on the right side of the line.  Then we draw our diagonal lines to see what we have to add and subtract.  The first three diagonal lines you are supposed to add  all the numbers together.  The diagonal lines going the opposite way, you have to subtract all those numbers together.  Then you get the two numbers that you added and subtracted and you subtract them together.  That should give you your determinate.  In this problem, the determinate that I found was -32.

Adding and Subtracting Matrices

Steps for adding and subtracting matrices. 

A^t means transpose.

When you transpose a matrix, you switch the rows with the columns and columns with the rows.  

Example: 

(6 7 8)

(8 0 2)

(2 8 1)

When transposed, it comes out as: 

(6 8 2)

(7 0 8)

(8 2 1)

The dimensions of a matrix is found by using rows by columns.

Example: 

The dimensions of the matrix in the first example would be 3 x 3. (There are 3 rows and 3 columns.)

When adding a matrix with another matrix, they must have the same dimensions.

When adding matrices, you have to add the corresponding entries with each other. 

Example:

(9 8)         (1 2)

(0 1)         (9 3)

A. What are the dimensions of the two matrices? 

Answer: 2x2 

B. Add the following matrices. 

Answer: (9 8) + (1 2) = (10 10)

             (0 1)    (9 3)    (9    4)

Matrix Multiplication

I am coming do this blog now, so I can go finish all of your study guidess..

--To multiply matrices
(a)(b)x(b)(c)
The inner numbers have to match

--The resulting matrix will be a(c)

Example 1:
Find each matrix product, if it is defined.
[4 3 [5 = [23
-1-2] 1] -7]
2x2 2x1
4(5)+3(1)=23
-1(5)+-2(1)=-7

Example 2:
Find each matrix product, if it is defined.
[-6 [-1 12 = Not Defined
0] 0 -4]
2x1 2x2
*inner numbers do not match, so it is not defined

Example 3:
Find each matrix product, if it is defined.
[1 -5 [4 -4 = [4 -9
2 3] 0 1] 8 -5]
2x2 2x2
1(4)+-5(0)=4
1(-4)+-5(1)=-9
2(4)+3(0)=8
2(-4)+3(1)=-5

Example 4:
Find each matrix product, if it is defined.
[-2 3 [0 3 = [-18 9
4 2] -6 5] -12 32]
2x2 2x2
-2(0)+3(-6)=-18
-2(3)+3(5)=9
4(0)+2(-6)=-12
4(3)+4(5)=32

Example 5:
Find each matrix product, if it is defined.
[8 -10 [-2 = Not Defined
0 3 -9
-6 4] 1]
3x2 3x1
*inner number do not match, so it is not defined

***Again, I am sorry for the messed up matrices. I haven't figured out how to fix them yet.

Cramer's Rule

x = Dx/D y = Dy/D z = Dz/D

Where D is the determinant of the coefficient matrix

Used to solve systems of equations using matrices and determinants

Ex. 1 3x - y + 2z = 4
2x + 3y - z = 14
7x - 4y + 3z = -4

x y z x y z
[ 3 -1 2 ] [ 4 -1 2 ]
D = [ 2 3 -1 ] = -30 Dx = [ 14 3 -1 ] = -30
[ 7 -4 3 ] [ -4 4 3 ]

x y z x y z
[ 3 4 2 ] [3 -1 4 ]
Dy = [ 2 14 -1 ] = -150 Dz = [ 2 3 14 ] = -90
[ 7 -4 3 ] [ 7 4 -4 ]

x = -30/-30 = 1 y = -150/-30 = 5 z = -90/-30 = 3

Ex. 2 7x + 4y = 19
3x - 10y = 14

D = [ 7 4 ] = -82 Dx = [ 19 4 ] = -246
[ 3 -10 ] [ 14 -10 ]

Dy = [ 7 19 ] = 41 x = -246/-82 = 3 y = 41/-82 = -1/2
[ 3 14 ]

Adding/subtracting matrices

Chapter 14 Section 1- Adding and subtracting matrices.

Steps to adding and subtracting matrices:

1)A^t, to transpose : switch rows and columns.
2)In order to add matrices they have to have the same dimensions.(rows x columns)
3)In order to add matrices you must add corresponding entries.

Example 1. What are the dimensions of B?
[2 8]
[8 5]
[7 2]
[5 3]

Answer: 4 x 2

Example 2. Add D and B
D=[2 4] C= [ 2 10]
[16 1] [6 2]
[9 13] [1 3]

Answer: [4 14]
[22 3]
[10 16]

Matrices :)

Matrices-adding and subtracting

A^t (transpose) means switching rows and columns.
To add matrices they must have the same dimensions.(rows X columns)
To add matrices you will add corresponding entries.

Ex.1:
Find dimensions by counting how many rows there are and how many columns:
[145]
[783]
2X3

Ex.2:
Find the dimensions of D:
There are three rows and also three columns therefore the dimensions would be three by three.
[0 9 -3]
[7 -4 8]
[11 2 3]
3X3

Ex.3:
To do this you will subtract all the numbers of Matrice A to Matrice B and you will then get a new matrice.
Find A-B
A=[0 9 -3] [7 5 4]
[7 -4 8] - [2 1 8] = [7 4 -7]
[5-5 0]

Ex.4:
To find this you will add the numbers from matrice A to matrice B and you will get a new matrice.
Find A+B:

A=[0 9 -3] [7 5 4] = [7 14 1]
[7 -4 8] + [2 1 8] [9 -3 16]

Matrices

A^t - transpose – switch rows and columns

To add matrices they must be the same dimensions(row x column)

To add matrices you add corresponding entries

To complete scalar multiplication you multiply every entry by the number

Ex 1. A = [ 3 8 1 ] B = [ 2 0 9 ]

[ 4 0 -3 ] [ 4 -6 -5 ]

[ -2 1 5 ] [ 0 7 2 ]

a. What are the dimensions of A and B? A = 3x3 B = 3x3

b. Find A^t and B^t A^t = [ 3 4 -2 ] B^t = [ 2 4 0 ]

[ 8 0 1 ] [ 0 -6 7 ]

[ 1 -3 5 ] [ 9 -5 2 ]

c. Find A + B A + B = [ 5 8 10 ]

[ 8 -6 -8 ]

[ -2 8 7 ]

d. Find 4A – 2B 4A = [ 12 32 4 ] 2B = [ 4 0 18 ] 4A – 2B = [ 8 32 -14 ]

[ 16 24 -12 ] [ 8 -12 -10 ] [ 8 12 -2 ]

[ -8 4 20 ] [ 0 14 4 ] [ -8 -10 16 ]

14-1 :-) KAYLA

14-1 is adding and subtracting matrices! :)
-A^t (transpose)= switching rows and columns.
-To add matrices theyhave to have the same dimensions.(rows x columns)
-To add matrices you have to add corresponding entries.

Example 1.) what are the dimensions of K.
[0]
[1]
[11]
[10]

ANSWER: (4 x 1)

Example 2.) add K and C.
K=[1 1] C= [ 6 9]
[11 10] [3 13]
[10 26] [10 2]

ANSWER: [7 10]
[14 23]
[20 28]

14-1!

Adding and Subtracting Matrices.

~A^t (transpose)= switching rows and columns.

~To add matrices they must have the same dimensions.(rows X columns)

~To add matrices you will add corresponding entries.

_____________________________________________________________________

Ex.1:

Find the dimensions of Z:

[234]

[125]

2X3

Ex.2:

Find the dimensions of H:

[0 9 -3 2]

[7 -4 8 6]

[11 2 3 0]

3X4

Ex.3:

Find A+B:

A= [2 8] B= [11 -3]

[5 6] [-5 15]

[13 5]

[0 21]

Ex.4:

Find X-Y:

X=[21 6] Y=[0 13]

[15 3] [-7 0]

[21 -7]

[22 3]

Adding and Subtracting Matrices

Matrix Addition and Subtraction

In order to add or subtract matrices, the dimensions of each matrix must be the same.  If the dimensions are not the same, the sum or difference is undefined.  Then, you add or subtract the matching numbers in the two matrices.

1.      [2   -4    6        +    [9    -3     -8       =   [11    -7     -2

4      9    -7]            0     -2      5]             4      7      -2]

2.      [ 4     -12     5        -     [-4      -5     -13               =      [8     -7     18

-2       -34     3]               -3      15     12]                      1     -49     -9]

Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must be the same as the number of rows in the second in order to multiply.  For example a 2x3 matrix and a 3x4 matrix can be multiplied because the inside numbers are the same.

Once you know you can multiply, you then multiply each number in the first row by each number in the first column and add the products together.  You do this for each row and each column.

1.      [-2     5     -3                      [-2      5

1       -4      -2]        x            5     -2

                                             -3     -1]

This is a 2x3 and 3x2, so you can multiply.

=[(-2)(-2) + (5)(5) + (-3)(-3)                      (-2)(5) + (5)(-2) + (-3)(-1)

    (1)(-2) + (-4)(5) + (-2)(-3)                      (1)(5) + (-4)(-2) + (-2)(-1)]

=[ 4+25+9                 -10-10+3

    -2-20+6                   5+8+2]

=[38       -17

   -16       15]

Love, Jenna

Matrices

A^t means transpose the matrix. To do this, flip the columns and rows.

To add matrices, both matrices must be the same dimensions(rows by columns).

To multiply matrices, the dimensions of both matrices must match like so: (Matrix 1: A x B, Matrix 2: B x C. If both B's match, dimensions of resulting matrix will be A x C).

Multiplying matrices is too complicated to explain with words, so I'll just show you.

Ex 1.

A = [ 3 8 1 ] B = [ 2 0 9 ]

.......[ 4 0 -3 ] [ 4 -6 -5 ]

.......[ -2 1 5 ] [ 0 7 2 ]

Dimensions are 3 x 3.

Find A^t and B^t.

A^t = [ 3 4 -2] B^t = [ 2 4 0 ]

...........[ 8 0 1 ] [ 0 -6 7 ]

...........[ 1 -3 5 ] [ 9 -5 2 ]

Find A + B

A + B = [ 5 8 10]

..............[ 8 -6 -8 ]

..............[ -2 8 7 ]

^^^Everything you need for matrices.

Matrices

-A^t, which means transpose,=switch rows and columns.

-You add corresponding entries when adding matrices.

-You multiply every entry by the number to complete scalar multiplication.

Solve.

1. [-3 + [1 = [-2

0] 2] 2]

2. [12 7] + [3 -4] = [15 3]

3. [8 1 - [3 -2 = [5 3

-1 5] 4 -1 ] -5 6 ]

4. [1 -1 0] - [6 -9 -1]= [-5 8 1]

5. [8 2 -2 + [12 3 10 = [20 5 8

-3 1 14] 0 0 -6 ] -3 1 8 ]

Matrices!

I was very proud of myself withthis lesson because I actually knew what I was doing! :)

*A^t(transpose)=switch rows and columns
*To add matrices they must be the same dimensions (rows and columns).
*To add matrices, you add cooresponding entries.
*To complete scalar multiplication, you multiply every entry by the number.

Example 1:
What are the dimensions of M?
M= [1 8 9
6 4 0]
2x3

Example 2:
What are the dimensions of M?
M=[ 8 9 6 5
0 8 4 7
2 9 4 6]
3x4

Example 3:
Find A + B
A= [-3 B=[1 A+B=[-2
0] 2] 2]

Example 4:
Subtract
[8 1 - [3 -2 = [5 3
-1 5] 4 -1] -5 6]

Example 5:
Add
[8 2 -2 + [12 3 10 = [20 5 8
-3 1 14] 0 0 -6] -3 1 8]

Example 6:
Multiply
8[5 -2 = [40 -16
4 0] 32 0]

Example 7:
Multiply then Add
2[3 0 + [-2 -2 = [6 0 + [-2 -2 = [4 -2
-4 1 3 0 -8 2 3 0 -5 2
0 -1] 6 11] 0 -2] 6 11] 6 9]

BTW, I am sorry for all of the messed up matrices, I can't figure out how to fix them.

Matrices!

This week we learned Matrices!
-A^t, the t meaning transposed with means to switch rows and columns.
-To add matrices they must be the smae dimensions (rows x columns)
-To add matrices you add coresponding entries
-To complete scalar multiplication you multiply every entry by the number.

Example 1:
3 8 1 =A 2 0 9 =B
4 0 -3 4 -6 -5
-2 1 5 0 7 2

a. What are the dimensions of A and B
A= 3x3
B= 3x3

b. Find A^t and B^t
A^t= 3 4 -2 B^t= 2 4 0
8 0 1 0 -6 7
1 -3 5 9 -5 2

c. Find A+B
3+2 8+0 1+9
4+4 0+(-6) +(-3)+(-5)
(-2)+0 +1+7 +5+2

= 5 8 10
8 -6 -8
-2 8 7

d. Find 4A-2B
Because im lazy I used my calculator, but first you multipy A by 4 and B by 2 then you subtract them.

= 8 32 -14
8 12 -2
-8 -10 16

Hope ya"ll had a good weekend :)
--Danielle

11-2

9cis4pie/3

y=9sin240 degrees                       x=9cos240 degrees
y=-7.794                                     x=-4.5
z=-7.794+-4.5i

We are given 9cis4pie/3 and we have to solve it.  To solve this problem, you have to separate the cis into sin and cos.  Then you have to figure out what 4pie/3 is in degrees to make the problem easier.  To separate the cis, you have to do y=9sin240degrees and x=9cos240 degrees.  Then you just plug those two into your calculator and you get that y equals -7.794 and x equals -4.5.  Then to solve for z you get z equals -7.794+-4.5.

Matricies

Matricies are groups of numbers that are done with a special set of rules. Matricies are usually in a set of large parenthesis. There are a few rules to solving matricies.

-Tranpose (means to switch rows and columns). Represented by a little floating t.
-To add matricies they must be the same dimensions. The deminsions of a matrix are (row * columns)
-To add matricies you add corresponding entries
-To complete scalar multiplication you multiply ever entry by the number. In other words multiply every number by the number you have to muliply by.

Ex1.Find the demensions of the following matrix then transpose.
{2 3 1}
{0 4 -1}
A.The demensions are 2*3. Two rows by three columns.
B.Transposed is {2 0}
{3 4}
{1 -1}
The demensions usually change once transposed. New demensions are 3*2

Ex2.Multiply the following matrix by 3.
{5 0}
{3 2}
A.Multiply each number by 3. {15 0}
{9 6}
Demensions do not change when multiplied by single number.

Matrices

A matrix is an array of numbers.

To show a matrix's size, you put the rows x columns.

To add two matrices, just add the numbers in the matching positions.

The two matrices must the be same size.  They must have the same amount of columns and rows to match in size.

Transposing a matrix is when you swap the rows and columns.

A^t is the symbol for transposing.

Example 1

A=(2 -3)    B=(-1 -5)
     (-4 2)          (3 -2)

A. A^t= (2 -4)   B^t= (-1 3)
              (-3 2)            (-5 2)

B. A+B=(1 -8)
               (-7 0)

   

11-1

11-1 Polar and Rectangular

Rectangular Form is stated as (x,y)
Polar Form is stated as (r,y)

To convert from polar to rectangular use the formulas:
~y=rsiny
~x=rcosy

To convert from rectangular to polar use the formulas:
~r=sqrt. x^+y^2
~y=tan^-1(y/x)
________________________________________________________

Ex.1: Give the rectangular coordinates for (4,60)
You plug these numbers into the formula
y=4sin60
x=4cos60
Usually you use the trig chart to figure these out,or you can use your calculator
y=4(.5)
x=4(.866)
~Multiply the equations.
y=2
x=3.464
~Put in point form.
(3.464,2)

Ex.2: Find the polar coordinates of (5,6)
Plug numbers into the formula
r= the square root of x^2 + y^2
r= the square root of 5^2 + 6^2 =7.810

y=tan^-1(6/5)
~Solve the two equations.
r=8
y=50.194
~Put into two points.
(8,50.194)
(-8,50.194)

Plane Curves

Plane Curves... yeah.

So the different types of curves we need to know are:

1. Lemniscate
2. Cardioid
3. Rose
4. Limascon of Pascal
5. Spiral of Archimedes

For the test, you will need to know what each one looks like, and how the equation of each one looks like.

A Lemniscate looks like:
and it's equation will either look like...

1. r² = a²cos2Θ
2. r = ± √(a²cos2Θ)
Ex. r² = 9cos2Θ
r = ± √(9cos2Θ)

A Cardioid resembles..a heart..kinda.
It's equation will look like ---> r = a + acosΘ.
Ex. r = 3 + 3cosΘ
r = 5 + 5cosΘ

A rose will obviously look like a flower of sorts.
The equation will look like: r = asinbΘ or r = acosbΘ

To find the number of petals, if b is odd, it has b petals. If b is even, it has 2b petals.

Ex. 3sin8Θ has 16 petals

3cos9Θ has 9 petals

A Limascon of Pascal looks similar to a Cardioid:
the equations looks like: r = b + acosΘ
the difference between a Cardioid and a Limascon is that b is not equal to a.

To find if it has an inner loop or not, if a is greater than b, it does, if a is less than b, it does not.

Ex. 3 + 5cosΘ does have an inner loop

5 + 3 cosΘ does not have an inner loop


A Spiral of Archimedes is a spiral, obviously :P
LOOK HOW PRETTY!! Haha, uh, the equation is pretty much r = aΘ

Ex. r = 8Θ