-8 5

2 1 By: Veronica Chiang LEARNING GOALS What is a "Matrix"? Adding and Subtracting Matrices History of The Matrix Questions Multiplying a Matrix by a Constant Thank you for listening! To understand and know the purposes and characteristics of a matrix

To know a little bit about the history of the matrix

To know some different types of matrices

To be able to add, subtract, and multiply matrices

To be able to transpose matrices

To be able to find the determinant of a square matrix

To know how matrices can be used in real life (application of matrices) According to the dictionary, a matrix is:

1. An environment or material in which

something develops.

2. A mass of fine-grained rock in which gems,

crystals, or fossils are embedded. To add or subtract any matrix, both of them must be the same size or have identical dimentions.

That is, the number of rows and columns of the matrix must match in size. To multiply a matrix with a constant value, you have to multiply the constant with each of the numbers in the matrix. The history of matrices dates back all the way to 200 BC, when Chinese mathematicians use arrays of numbers. However, the term "matrix" was not applied to the concept until 1850 by James Joseph Sylvester, an English mathemetician. Matrices But in mathematics, a matrix (plural matrices) is defined as a way of representing numbers in a rectangular array. The numbers, known as ELEMENTS or an ENTRY, are organized into rows and columns and are held by a single bracket. A single matrix is usually represented by a capital letter and classified by their dimension

(# of rows x # of columns).

Matrices are a strand of linear algebra. 1 2

3 4 Therefore, these are are matrices When you search up the word Matrix, you will most likely get the name of a 1999 movie called "The Matrix". 22 -5 10

-36 19 53 a a a

a a a

a a a 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 Multiplying a Matrix by Another Matrix THE END Questions! The word "matrix" comes from the Latin word meaning "womb", and it retains that sense in English. Generally, it can also mean any place in which something is formed or produced. Other important people who contributed and revolutionized the matrix included Arthur Cayley, Georg Frobenius and Werner Heisenburg. Important: Addition Example: To add two matrices, add the numbers in the matching positions. These are the calculations:

3+4=7 8+0=8

4+1=5 6+(-9)=-3 Subtraction Like addition, to subtract two matrices, subtract the numbers in the matching positions. Example:

These are the calculations:

3-4=-1 8-0=8

4-1=3 6-(-9)=15 Subtracting is actually defined as the addition of a negative matrix: A + (-B) Add 8 3

0 4 [ ] + -2 9

0 7 [ ] = Subtract -5 1

6 14 -9 3

4 20 [ ] [ ] - = Answers Add 8 3

0 4 -2 9

0 7 6 12

0 11 [ ] [ ] + = Subtract -5 1

6 14 -9 3

4 20 [ ] [ ] - = 4 -2

2 -6 Example: constant These are the calculations:

2x4=8 2x0=0

2x1=2 2x(-9)=-18 We call the constant a scalar, so this is called "scalar multiplication". Row Matrix

A row matrix is formed by a SINGLE row. Types of Matrices There are many different types of matrices but these are the most common ones: 89 -31 55 24

9

76 Rectangular Matrix

A rectangular matrix is formed by a different number of rows and columns. Column Matrix

A column matrix is formed by a SINGLE column. 6 7 2

5 8 5 Square Matrix

A square matrix is formed by the same number of rows and columns. This is a 2 by 3 matrix.

It has 2 rows and 3 columns Identity Matrix

An identity matrix is a type of square matrix in which the diagonal elements are equal to 1 and the rest of the elements are equal to 0. It is like the number "1" in multiplication of matrices. -2 3 1

-8 4 2

9 7 -5 3x3 Matrix 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1 How to Transpose a Matrix To "transpose" a matrix, just swap the rows and columns. You must write a "T" in the top-right corner of the matrix to indicate "transpose". Example #1: 6 4 24

1 -9 8 T = 6 1

4 -9

24 8 Example #2: 81 25 4

7 6 -21

-33 1 99 = T 1st Column becomes the 1st Row

2nd Column becomes the 2nd Row

3rd Column becomes the 3rd Row Transpose this Matrix 81 7 -33

25 6 1

4 -21 99 HINT -91 75 -4 1

87 -5 23 88

2 19 -6 -54 A = A = ? T -91 75 -4 1

87 -5 23 88

2 19 -6 -54 Answer T = -91 87 2

75 -5 19

-4 23 -6

1 88 -54 Multiplying matrices is more complicated but I will show you how to do it with two simple 2 by 2 matrices, step by step. 1 2

3 4 5 6

7 8 x Matrix multiplication is not like addition or subtraction!

You do not multiply 1 by 5, 2 by 6, 3 by 7, and 4 by 8 1 2

3 4 x = 5 6

7 8 To get the first element of the matrix (number that is in the first column of the first row, in this case "w"):

You use the first row of numbers in the first matrix and the first column of numbers in the second matrix

Multiply the 1st numbers (1x5) and the 2nd numbers (2x7) and add them together.

1x5+2x7 = 5+14 = 19

w = 19 w x

y z To get the second element of the matrix (number that is in the second column of the first row, in this case "x"):

You use the first row of numbers in the first matrix and the second column of numbers in the second matrix

Multiply the 1st numbers (1x6) and the 2nd numbers (2x8) and add them together.

1x6+2x8 = 6+16 = 22

x = 22 (Step 1) (Step 2) 1 2

3 4 x = 5 6

7 8 w x

y z To get the third element of the matrix (number that is in the first column of the second row, in this case "y"):

You use the second row of numbers in the first matrix and the first column of numbers in the second matrix

Multiply the 1st numbers (3x5) and the 2nd numbers (4x7) and add them together.

3x5+4x7 = 15+28 = 43

y = 43 1 2

3 4 x = 5 6

7 8 w x

y z (Step 3) 1 2

3 4 x = 5 6

7 8 19 22

43 50 (Step 4) To get the fourth element of the matrix (number that is in the second column of the second row, in this case "z"):

You use the second row of numbers in the first matrix and the second column of numbers in the second matrix

Multiply the 1st numbers (3x6) and the 2nd numbers (4x8) and add them together.

3x6+4x8 = 18+32 = 50

z = 50 1 2

3 4 x = 5 6

7 8 w x

y z Answer Determinants The determinant can only be found in square matrices. They are whole numbers associated with that matrix and are most often used to find the nature of solution of the system of linear equations defined by the matrix. A = B = C = J.J. Sylvester To find the determinant of a 2×2 matrix:

First you multiply the diagonals and then subtract the bottom number from the top number. Example: A = Step 1: Multiply diagonal numbers 2 x 5 = 10

(-8) x 1 = -8 Step 2: Subtract (bottom-top) -8 5

2 1 *Note* When finding the determinant, the square brackets become lines (-8) - 10 = -18 The determinant of this matrix is -18 -8 5

2 1 2 x 5 = 10

(-8) x 1 = -8 Find the Determinant of this Matrix: A = 9 -1

-1 2 Answer 9 -1

-1 2 = (-1) x (-1) = 1

9 x 2 = 18

18 - 1 = 17 Cayley Frobenius Heisenburg Questions Multiply 3 x -8 4

-1 3 [ ] = 5 1

-2 2 9 -3

-6 7 [ ] [ ] x = Answers 3 x -8 4

-1 3 [ ] = [ ] To multiply, matrices must have the same “inner dimensions"

i.e. (4x3) x (3x2)

The final answer will have the “outer dimensions” of your matrices

i.e. (4x3) x (3x2) 3 and 3 are the inner dimensions column # in 1st matrix = row # in 2nd matrix your matrix will have a dimension of 4x2 (4 rows and 2 columns) -24 12

-3 9 5 1

-2 2 x [ ] 9 -3

-6 7 = Order is important because of row by column, so A×B is different from B×A 39 -8

-30 20 Get ready... Question #1 In the mathematical world, what is a matrix (characteristics of a matrix)? Question #2 What is an identity matrix?

What happens when you multiply an identity matrix with another matrix? Question #3 Solve: -34 95

101 -77 91 2

-154 -12 + Question #4 What do you call the number inside a bracket of a matrix? Question #7 BONUS 63 -62 1

-7 -97 35

56 -48 89

4 14 24 -2 4

1 -5

0 3 x Solve: Question #6 Question #5 Find the determinant of this matrix: A = 4 7

-1 6 If A = and B = 2 1

5 -3 -2 1

3 -2 What is 2A + B ? T 4 and 2 are the outer dimensions row # in 1st matrix and column # in 2nd matrix can vary You use the transpose operation to help determine the inverse of a matrix. This is needed when you want to solve a given set of linear equations of several unknowns. MINUS - Matrices are useful when dealing with many numbers at once because all the numbers can be integrated into just one calculation using matrix algebra. An example of how matrices can be used is with encryption. When programmers encrypt a message, they can use matrices and their inverse. The internet could not function without encryption, and neither could banks since they now use these same methods to transmit private and sensitive data. Matrices are also sometimes used in computer animation.

Another application of matrices would be for geologists for seismic surveys (studies done to gather and record patterns of induced shock wave reflections from underground layers of rock). Matrices are also used in graphs and statistics for doing scientific studies in many other different fields. They are used in calculating gross domestic product in economics more efficiently since you can work with many numbers at once. Application of Matrices in Real Life RULES FOR MULTIPLICATION: You have to multiply the row by column.

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# Matrices

Grade 8 - Math (Friday May 31, 2013)

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