Chapter+4

media type="file" key="Math Video.mov" Our awesome video!!!  =**Chapter 4- Matrices & Determinants** =  Note: I tried to make all the matrices line up, but I didn't have much luck. The | | symbols are the brackets. For some reason, I couldn't use the Tab key when I was editing it, and when I used spaces to indent the lines to line them up, the extra spaces got deleted when it saved. I couldn't figure out a way around it, so I tried the best I could. Sorry! -Liz
 * __**Roles of Wiki**__ || __**Student #1**__ || **__Student #2__** ||
 * Editor || Devin Gohn || //put your first initial and last name// ||
 * **Multimedia** || Chloe Nagle ||  ||
 * **Text** || Liz Bishop! ||  ||
 * **Practice Problems** || crystal murphy ||  ||
 * **Career/Real Life App** || W. Bryan S. ||  ||

=//**4.1: Matrices**// =
 * Matrix**- a rectangular arrangement of numbers in rows and columns (rows are horizontal & columns are vertical)
 * Dimensions**- (rows x columns)
 * Entries**- numbers inside the brackets
 * Column Matrix-**
 * 2| (3 x 1)
 * 1|
 * 5|


 * Square Matrix-**
 * 2 1| (2 x 2)
 * 7 3|


 * Zero Matrix-**
 * 0 0|
 * 0 0|

=
 * Equal Matrix**- equal entries in same positions
 * Ex:**
 * 1/2 9/3|
 * 100% .75|
 * .5 3|
 * 1 3/4|


 * Scalars**- number in front of matrix which means to multiply (distribute)

=**//4.2: Multiplying with Matrices//** = [B] (n x m)** In order to multiply matrices [A]*[B], A's column must match B's rows. It's called **defined** when that is true. The product's dimensions will be the number of A's rows & B's columns. To multiply, you first multiply the 1st row in the first matrix by the 1st column in the second matrix. Add the products, and the sum goes into the 1st row, 1st column spot of the matrix in the answer. Next, Multiply the 1st row by the 2nd column, then the 2nd row by the 1st column, etc. Multiply the bold numbers & add the products together. -2(-1) + 3(-2) = -4 Product:
 * [A]*[B] Dimensions: [A] (m x n)
 * Ex:**
 * **-2** **3** | | **-1** 3 | [Dimensions are (3 x 2) & (2 x 2), so the product's dimension will be (3 x 2).]
 * 1 -4 | * | **-2** 4 |
 * 6 0 |
 * -4 _ |
 * _ _ |
 * _ _ |

Product:
 * **-2 3** | | -1 **3** |
 * 1 -4 | * | -2 **4** | Multiply the bold numbers & add the products together. -2(3) + 3(4) = 6
 * 6 0 |
 * -4 6 |
 * _ _ |
 * _ _ |

=**//4.3: Finding Determinants of Square Matrices//** = = det. = ad - bc
 * (2 x 2):**
 * a b |
 * c d |
 * a b |
 * c d |

= det 2(-1) - (-3)(4) det det = (aei + bfg + cdh) - (ceg + afh + bdi)
 * Ex:**
 * 2 -3 | | 2 -3 |
 * 4 -1 |
 * 4 -1 |
 * 2 -3 |
 * 10**
 * (3 x 3):**
 * a b c | --> | a b c | a b det = (aei + bfg + cdh) - (ceg + afh + bdi)
 * d e f | --> | d e f | d e
 * g h i | --> | g h i | g h

det det = (-56 + 0 + (-10)) - (-7 + 0 + 30) = **-89**
 * Ex:**
 * 4 3 1 | 4 3
 * 5 -7 0 | 5 -7
 * 1 -2 2 | 1 -2

Area of a Triangle: A = +/- .5(det) - A must be positive. - Order of coordinates when finding the determinants doesn't matter. - Make the matrix (3 x 3) by adding an extra column of 1's as the third column. //First find the det.// det = -22 //Next put it into the formula for the area of a triangle. Since the determinant is negative, and an area of a figure must be positive, multiply it by -.5 to make it positive.// A = -.5( -22 ) A = **11 square units**
 * Ex:** Find the area of the triangle with vertices at A(0, 2), B(2, 8), and C(5, 6).
 * 0 2 1 | 0 2 det = -22
 * 2 8 1 | 2 8
 * 5 6 1 | 5 6

=//**4.4: Inverses & Identities**// = Main diagonal and 1’s & other entries are 0’s.
 * (2 x 2):** I =
 * 1 0 |
 * 0 1 |
 * (3 x 3):** I =
 * 1 0 0 |
 * 0 1 0 |
 * 0 0 1 |

A B A^(-1) means inverse.
 * 3 -1 |
 * -5 2 |
 * 2 1 |
 * 5 3 |
 * If the dimensions are the same, they’re commutative. (AB = BA)

//**Inverse of (2 x 2):**// A =
 * a b |
 * c d |

A^(-1) = __1__ | d –b | (If ad – cb doesn’t = 0.)
 * A| | -c a |

= = =**//4.5: Solving Equations with Matrices//** = 2x = 6 (1/2)2x = (1/2)6 x = 3 [A][X] = [B] [A^(-1)] [A] [X] = [A^(-1)] [B] (*Remember the order!) [X] = [A^(-1)] [B] <Order to solve matrix equations. if A = and B = Answer:
 * Algebra example:**
 * Matrix example:**
 * Ex:** (2 x 2) Solve for AX = B
 * 4 -1 |
 * -3 1 |
 * 8 -5 |
 * -6 3 |
 * Steps-**
 * 1.** Find determinant of [A]
 * 2.** Multiply by the reciprocal of the det if det doesn't = 0.
 * 3.** Be sure you changed [A] the a & d positions and the b & c signs.
 * 4.** Multiply [A^(-1)] [B] = [X] in the ORDER
 * 2 -2 |
 * 0 -3 |

(3 x 3) inverse matrix: //**Graphing calculator steps-**// Enter this matrix: Answer:
 * 1.** 2nd matrix
 * 2.** arrow over to edit
 * 3.** press enter
 * 4.** set dimensions of matrix [A] to a (3 x 3)
 * 5.** enter entries, pressing enter after each entry
 * 1 -1 0 |
 * 1 0 -1 |
 * 6 -2 -3 |
 * 6.** 2nd quit. (home screen)
 * 7.** 2nd matrix
 * 8.** press 1 or enter
 * 9.** press "x^(-1) button below math key
 * 10.** press enter
 * -2 -3 1 |
 * -3 -3 1 |
 * -2 -4 1 |

2x + 3y + z = -1 3x + 3y + z = 1 2x + 4y + z = -2 Solve for ( x, y, z) Variable matrix: Constant: First enter a matrix under edit. Then arrow over to math, press "1" or enter. You should see "det( " on the home screen. You need to tell it which matrix. Press 2nd matrix, then press the number of the matrix which you are finding the determinant.
 * //System of Equations using Matrices://**
 * Step 1:** [A] = coefficient matrix:
 * 2 3 1 |
 * 3 3 1 |
 * 2 4 1 |
 * x |
 * y |
 * z |
 * -1 |
 * 1 |
 * -2 |
 * Step 2:** Find [A^(-1)]:
 * -1 1 0 |
 * -1 0 1 |
 * 6 -2 -3 |
 * Step 3:** Take [A^(-1)] times [B] to get the x, y, and z matrix. You can do this on the home screen of your calculator by typing matrix [B] into the calculator.
 * Step 4:** Press 2nd matrix. Arrow over to edit. Arrow down to "2". Press enter. Enter (3 x 1) dimensions for [B[. Enter entries.
 * Step 5:** Press 2nd mode (quit)
 * Step 6:** (on home screen) press "1", press key "x^(-1)", times [B], enter. You should get the values of x, y, and z as x = 2, y = -1, and z = -2. You can subst. these values into the system to see that they work.
 * //Finding the det. using the graphing calculator://**

Real Life Application!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1one!!!!!!

The Chinese studied matrices during 200 and 100 B.C. "There are 3 types of corn, of which 3 bundles of the 1st, 2 of the 2nd and 1 of the 3rd make 39 measures. 2 of the 1st, 3 of the 2nd and 3 of the 3rd make 26 measures. How many measures of corn are contained of one bundle of each type?" Their matrix is a 4x3. It has four rows and three columns and it is called a counting block" 1 2 3 2 3 2 3 1 1 26 34 39

**Example Problems!**

| 3 | | 1 | | 3+1 | | 4 |
 * example 1 **
 * 4 | + | 0 | = | 4+0 | = | 4 |
 * 7 | | 3 | | 7+3 | |10|

| 3 | | 1 | | 3-1 | | 2 | **example 3** | 2 0 | | 3(2) 3(0) | | 6 0 | 3 | 4 7 | = | 3(4) 3(7) | = | 12 21 |  | -2 3 |
 * <span style="color: rgb(0, 128, 0);">example 2 **
 * 4 | -- | 0 | = | 4-0 | = | 4 |
 * 7 | | 3 | | 7-3 | | 4 |
 * 4 5 | | 3(4) 3(5) | | 12 15 |
 * example 4**
 * 1 -4 | <span style="color: rgb(0, 0, 255);"> |-1 3 |
 * 6 0 | |-2 4 |

= | (-<span style="color: rgb(255, 0, 0);">2 )(-<span style="color: rgb(0, 0, 255);">1 ) + (<span style="color: rgb(255, 0, 0);">3 )(-2) (-2)(3) + (3)(4) |
 * (1)(-1) + (-4)(-2) (1)(3) + (-4)(4) |
 * (6)(-1) + (0)(-2) (6)(3) + (0)(4) |

= | -4 6 |
 * 7 -13|
 * -6 18 |