Essay On Matrices

Comprehensive Portfolio Project

Alex Abel

Table of Contents

Title 1

Table of Contents 2

Matrices 3

Solving Systems of Equations 4

Solving Systems of Equations Cont. 5

Matrices Examples 6

Matrices Examples Cont. 7

Set Theory 8

Set Theory Examples 9

Equations 10

Equations 11

Equation Examples 12

Functions 13

Functions Cont. 14

Function Examples 15

Function Examples Cont. 16

Matrices

A matrix in mathematics is a rectangular array of mainly numbers that are arranged in rows and columns. All of the individual numbers in the matrix are called the elements or entries. Matrices go back to the 17th century. The beginning of matrices started when studying systems of linear equations because of matrices helping in the solutions of those equations. Matrices were always known just as arrays back then. Matrices can be added, subtracted, and multiplied but with different rules. When adding and subtracting, matrices must be the same size in order to be solved. With multiplication, you must first find the dimensions and make sure that the inside numbers match. If they do, you then multiply each row by column. There are also three other ways to work with matrices: determinants, special multiplication, and inverses. For determinants, you have variables: A, B, C, and D. Remember: you can only find a determinant for square matrices, meaning 2x2. You will then put “switch and negate” into terms. You switch variables A and D, negate B and C, then subtract. After finding that information, you will put your determinant under 1 and solve. Special multiplication is just taking one ...

... middle of paper ...

... or odd, and positive or negative before you can determine your answer. Third, you have to see if your graph is above or below the x-axis between your x-intercepts and plug a value between these intercepts into your function. Last but not least, you plot your graph.

Function Examples

1.) Relation: {(1,4) , (8, 2) , (7, 3) , (9, 6)}

Domain: (1, 8, 7, 9)

Range: ( 4, 2, 3, 6)

2.) Relation: {(2, c) , (4, b) , (6, a)}

Domain Range

2 a

4 b

6 c

3.) f(x) = 4x2 + 8x + 3

-8 / 2(4) = -1

K = 4(-1)2 + 8(-1) + 3  4 – 8 + 3  K = -1

Vertex (-1, -1)

This problem’s arrows will go up because the first number in the equation is positive.

[ examples continued on the next page ]

0 = 4x2 + 8x + 3

M: 12

A: 8  (x-6) (x-2) = 0

X-intercepts  (6,0) (2,0)

Y = 4(0)2 + 8(0) + 3  Y-intercept = (0, 3)  [then you graph]

5.) f(x) = 4 – 2x2

Standard form: -2x2 + 4

Degree: 2