Period+8+Properties

=**Properties of Matrices**=

Kelsey Witt and Stacey Rosenfeld
>>>>> __Bibliography:__ http://www.purplemath.com/modules/matrices.htm http://www.purplemath.com/modules/mtrxmult2.htm http://chortle.ccsu.edu/vectorlessons/vmch16/vmch16_4.html http://www.purplemath.com/modules/mtrxmult3.htm
 * ===Size (order)===
 * The size of the matrices matters when multiplying. There has to be the same amount of columns in the seconds matrix as there are rows in the first matrix (the rows of A must have the same length as the columns of B).
 * Example 1:[[image:matrix_multiplication.png]]
 * If the rows and columns do not match up the correct way (mentioned above), the __**matrix is said to be undefined**__.
 * The size, also known as the order, of a matrix is given by the number of rows and columns it contains.
 * For example if we look here, we can see that there are 4 rows and 3 columns, therefore this would be referred to as a 4x3 (4 by 3) matrix.
 * [[image:matrix_size:order.png]]
 * Unlike usual multiplication __**order is important!**__Multiplication with matrices is not commutative. AB ≠ BA
 * For help with adding matrices
 * For help with multiplying matrices
 * ===Square Matrix===
 * When a matrix has the same number of rows as it does columns, m=n, it is known as a square matrix.
 * This is an example of a 3x3 square matrix.
 * [[image:square_matrix.png]]
 * Above, it says that AB ≠ BA, however in a square matrix, AB = BA <--- STACEY WE NEED TO EXPLAIN THIS
 * ===Column and Row Matrix===
 * The horizontal lines in matrices are known as rows (m) and the vertical lines are called columns (n).
 * A column matrix is a matrix that only has one column.
 * Matrix A = [[image:AV-17.png width="29" height="57"]]
 * Matrix A has only 1 column and 3 rows. As a result, it would qualify as a column matrix.
 * A row matrix is a matrix that only has one row.
 * Matrix B = [[image:AV-9.png width="71" height="32"]]
 * Matrix B has only 1 row and 3 columns. As a result, it would qualify as a row matrix.
 * ===Identity Matrix===
 * The Additive Identity
 * The identity property states that when zero is added to any real number, the number does not change. The number "0" is called the additive identity for real numbers
 * Here is a matrix denoted (0). When 0 is added to any matrix of the same dimensions, the matrix does not change:
 * [[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img342.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img343.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img344.gif]]+[[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img345.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img346.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img347.gif]]= [[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img342.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img343.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img344.gif]]
 * The Multiplicative Identity
 * The identity property states that when one is multiplied to any real number, the number stays the same (does not change). The number "1" is called the multiplicative identity for real numbers.
 * This is the identity matrix used for all matrices using the multiplicative identity:
 * I = [[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img7.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img8.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img9.gif]]
 * An example of this is:
 * [[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img30.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img31.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img32.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img33.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img34.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img35.gif]]=[[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img33.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img34.gif]][[image:http://img.sparknotes.com/figures/3/347c676afa6b7723d05bb2b4f2ace3c5/latex_img35.gif]]
 * **Extra Explanation-**
 * http://www.youtube.com/watch?v=CxPyD7ewuG4