Matrices - The Basics

Concepts:

A matrix is a very important mathematical object. Matrices (plural) have specific structures and functions that can be very useful in more complicated math. Though they may look intimidating at first, they are really not that complicated. You will need to understand how to do several mathematical functions with matrices. But before you can do that, you will first need to Understand the basic structure and properties of matrices. 

**INSERT LATEX OR PHOTOS OF MATRICES HERE TO USE AS A REFERENCE


_________________

Use This to Make Matrix:

\(
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\)

____________________

 

\(

\usepackage{amsmath}

\begin{equation}
B =
\begin{bmatrix}

a & b & c \\

d & e & f \\

g & h & i

\end{bmatrix}

\end{equation}

\)

Let’s start with some basic definitions and terminology. 

DEFINITION: An array of numbers arranged in rows and columns. 

   * Remember! Columns go up and down, rows go side to side

BASIC PROPERTIES OF A MATRIX:

   – ELEMENTS: The individual numbers that compose a matrix are called elements. 

   – DIMENSIONS: Every matrix has a certain number of rows and a certain number of columns. The convention is to say that a matrix is “\( m \) by \( n \)”, which means it has \( m \) rows and \( n \) columns.

   – POSITIONS: Every element has a unique position in the matrix. The convention is to refer to the position by saying is it in the \( i\)th row and the \( j \)th column.

TAKE-AWAYS!!

  – A matrix has \( m \) number of rows and \( n \) number of columns.

  – Every element has a unique position, the \( i\)th row and the \( j \)th column.

Key Concepts:

DEFINITION: A composite function (also known as a ‘composition of functions’) is when one function is found inside another. 

NOTATION AND TERMINOLOGY: Consider the composite function \( f(g(x)) \).  We read this as “f of g of x”. This composite function is the combination of two functions: \( g(x) \), which is the inner function, and \( f(x) \), which is the outer function.

Practice:

*PUT MATRIX HERE, ASK STUDENT TO IDENTIFY DIMENSIONS, CERTAIN ELEMENTS

\( g(x) = x^2 + 2x + 1 \)

\( f(x) = 3x \)

ANSWER: \( f(g(x)) = 3x^2 + 6x + 3 \)

Show your work!

  1. \( f(g(x)) = 3(g(x)) \)
  2. \( 3(g(x)) = 3(x^2 + 2x + 1) \)
  3. \( 3(g(x) = 3x^2 + 6x + 3 \)

What does \( f(f(x)) \) equal for \( x = 5 \)?

\( f(x) = 2x + 3 \)

ANSWER: \( g(0) = - 14 \)

Show your work!

  1. \(f(f(x)) = f(2x+3)\)
  2. \(f(2x+3) = 2(2x+3) + 3\)
  3. \(f(f(x)) = 4x + 9\)
  4. \(f(f(5)) = 4(5) + 9\)
  5. \(f(f(5)) = 29\)