Period+6+Matrix+Arithmetic+and+Operations

Addition/Subtraction- Jamie Scalar Multiplication- Jenna Transpose- Kelsey

__Addition/Subtraction__

Matrix addition is fairly simple. The __best__ way to teach the addition of matrices is by walking through an example problem.

Matrix subtraction is performed by adding the additive inverse of the matrix. In simpler terms, two matrices can be subtracted by finding the differences of the corresponding elements of the matrices (AM Textbook).
 * EXAMPLE 1:**

Please note that matrix addition AND matrix subtraction can only be performed if the involved matrices have the same dimensions (same number of rows and columns). Matrices of two different dimensions can never be equal.

Let A= Let B=
 * EXAMPLE 2:**

To Solve for A-B, simply subtract the corresponding numbers in the matrices:


 * Now, try this following example on your own:**



Here are 2 videos explaining how to add/subtract matrices (the first one is in a smaller dimension): http://www.youtube.com/watch?v=xhK-pzSJwJQ http://www.youtube.com/watch?v=xyAuNHPsq-g **(__Start__ at 5:25)**
 * Extra __Help__**

http://hmaricca.wikispaces.com/Period+6-Properties+of+Matrices
 * Also, if you would like to know exactly what a matrix is and/or what they are used for, please visit this wiki presented by Alex Volynsky and Caitlin Bakofsky:**

http://www.purplemath.com/modules/mtrxadd.htm http://stattrek.com/matrix-algebra/matrix-addition.aspx AM Textbook Youtube (Khan Academy)- http://www.khanacademy.org/
 * Sources:**

__Scalar Multiplication__ Scalar multiplication is multiplying a matrix by a number. To do scalar multiplication, simply multipy the number or scalar by each element of the matrix. *NOTE: Do not confuse scalar multiplication with multiplying matrices. Click here and use this wiki to review multiplying matrices and to see the differences between that and scalar multiplication.*

Before we begin, here is a helpful chart of the properties of scalar multiplication: Now that you know the properties of scalar multiplication, here are some basic examples:

As you can see in the example above, we multiplied the scalar, 2, by each element in the matrix.
 * EXAMPLE 1:**

Now that you know how to do scalar multiplication, let's move on to solving for a variable:

Given, find the value for x To solve this problem, all you have to do is find what number, when mulitplied by 2, results with 6. When x is mulitplied by 3, the result is 9. With x is multiplied by 5, the result is 15. And finally, when x is multiplied by 6, the result is 18. Set up one equation: 2x=6. So, **x=3**. When you plug in 3 for x, you will see that it works for all the elements in the matrix.
 * EXAMPLE 2:**

Now, try this more "complex" problem on your own where you have to solve for two variables: Find the values of j and f.
 * EXAMPLE 3:**

Solution: 2j-6=5 2(2)-f=3
 * j=5.5**
 * f=1**

If you need additional help with scalar multiplication, there are plenty of practice problems on the websites below. http://www.onlinemathlearning.com/matrix-scalar-multiplication.html http://www.purplemath.com/modules/mtrxmult.htm http://people.hofstra.edu/stefan_waner/realworld/tutorialsf1/frames3_1.html http://hotmath.com/hotmath_help/topics/scalar-multiplication-of-matrices.html
 * Sources:**

__Transpose__

Consider the dictionary definition of "transpose": With matrices, you are basically doing that same thing: changing the places of information given in a matrix. For example, the transpose of this matrix: is this: Instead of being a 2x4 matrix, this is now a 4x2 matrix.
 * Math Definition:** Information in a matrix can be given by interchanging the rows and columns of the matrix, E. Basically, Row # 1 in matrix E becomes column #1 in the transpose of matrix E. Click here to learn more about the columns vs. rows of a matrix.

-The transpose of the transpose of a matrix is just the matrix -The transpose of the sum of two matrices is the first transpose plus the second transpose of the matrices: -The transpose of the product of two matrices is the product of their transposes in reverse order : -The same holds true for the product of multiple matrices: -The transpose of a symmetric matrix is the same as the original matrix. Here's an example:
 * Notation:** A Transpose of a matrix is denoted: [[image:Screen_shot_2012-03-29_at_5.10.19_PM.png]] *E can be any letter, but t remains the same to represent "transpose"
 * Properties:**

Answer:
 * Problem 1:** [[image:Screen_shot_2012-04-06_at_3.22.47_PM.png]]Find [[image:Screen_shot_2012-04-06_at_3.24.05_PM.png]]

Find Answer:
 * Problem 2:**


 * Purpose:** Although it is not very difficult to find the transpose of a matrix, it can be very useful to do so. For example, instead of adding two matrices of dimensions 2x16, using the transpose of these, making them into 16x2 matrices would make them easier to add. *See above for more about adding matrices.

http://comp.uark.edu/~jjrencis/femur/Learning-Modules/Linear-Algebra/mtxdef/transpose/transpose.html AM textbook
 * Sources:**