9.6 The Algebra of Matrices

Hey guys!

So we're all pretty practiced when it comes to algebra and real numbers, but how about we focus on algebra of matrices. 

Really quick refresher: 

Matrix- one

Matrices- two or more

Elements- numbers within or that make up the matrix

The dimensions of a matrix are # of rows by #of columns

An element is identified by the row and column it is in

The Actual Thing

Addition of Matrices

- Two matrices can be added only if they have the same size

- Add the elements in corresponding positions of each matrix

Ex:

-The m x n zero matrix or additive identity is a matrix with all elements = 0

Such as: 

OR

OR

...you get the point

-The additive inverse of a matrix A is obtained by changing the sign of each nonzero element of matrix A

So:

The additive inverse of 

is

-Theorem on Matrix Properties

If A, B, and C are m x n matrices and if O is the m x n zero matrix

1.  A+B = B+A

2. A+ (B+C) = (A+B) + C

3. A+O = A

4. A + (-A) = O

Subtraction of Matrices

- Subtraction is pretty simple if you understand addition

we simply treat   A-B    as    A+(-B)

- just as in addition we subtract the elements in corresponding positions

Multiplication of Matrix

-When multiplying a real number  c  by a matrix A  you multiply each element of A by c 

EX:

100 200 300 400

=    A

                                                                                 5      =    C

5

100 200 300 400

=

5 * 100 5 * 200 5 * 300 5 * 400

=

500 1000 1500 2000

-Theorem On Matrix Properties

If A and B are m x n matrices and if c and d are real numbers, then

1. c(A+B) = cA + cB

2. (c+d)A = cA + cdA

3. (cd)A = c(dA)

Multiplication of Matrices

-When multiplying to matrices, the number of columns in the first matrix must = the number of rows in the second matrix

(remember dimensions of a matrix are rows x columns)

- Because of that

if the 1st matrix is 2 x 3 

and the 2nd matrix is 3 x 5

the resulting matrix will be 2 x 5

- When multiplying two matrices an element in row 1 column 1 is equal to row 1 of the first matrix multiplied by column 1 of the second matrix

So:

if

and

then 

- Something thats important to remember is that

The book gives a good explanation and example of this. and I'm getting pretty sick of typing everything up, so here

-If you still need help with Matrix Multiplication here is a website that explains it in yet another way, I found it very helpful and easy to understand:

http://www.mathsisfun.com/algebra/matrix-multiplying.html

and if you are more of an auditory learner heres a nice little video from Khan Academy

http://www.khanacademy.org/math/algebra/algebra-matrices/v/matrix-multiplication--part-1

actually its on the longer side but hey #firstworldstruggz
#ijustusedahashtag  #eww

-Almost done, okay multiplicative identities are always squared. They are also in reduced echelon form!
SO: the multiplicative identity for a 2x2 matrix would be

the multiplicative identity for a 3x3 would be 

And so on..

So that's what's up with Algebra of Matrices. 

People may doubt what you say

but they will believe what you do.

That's what the fortune cookie I just ate said. Deep. You can go ponder that now.

Oh and just cause I'm a pushy business woman, heres the link to my itunes: http://itunes.apple.com/us/album/leah-lavigne-ep/id480771601

-Leah