Algebra 2 Intro To Matrix Algebra
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The Matrix
Intro To Matrix Algebra

 

 There are other types of algebra

 besides the type we have been doing here.
 
 The most famous of these is called Matrix algebra.
 It is used a bunch in real life and other math.
 
 Because it is such a different system,
 the people who design math classes
 usually give you a taste of it now.
 That way it won't be brand new when you see it later.
 
 Here's how it works.
 
 Example:
 
 Say we are given these two equations to solve:
 

 3X - 5Y = 6     and     2X + Y = 5

 
 This one isn't too hard, 
 and we could solve it using the methods we already know.
 But we're going to use matrix algebra
 so you can see how it works.
 
 The first step is to write the two equations like this:
 
  3X - 5Y = 6
  2X + Y = 5
 
 Now write just the X and Y coefficients,
 arranged exactly the same way.
 
  3   5
  2   1
 
 Now we are going to combine these 4 numbers
 in a very specific way.
 
 First we multiply the top left 
 and bottom right numbers
 

 

 
 Then multiply the bottom left and top right numbers ...
 

 

 
 Then take the first multiplication
 and subtract the second multiplication ...
 

 3 - (-10) = 13

 
 Let's make sure you understand what we just did ...
 
 Here's one of those number groups (called a matrix)
 with letters in it instead of numbers:
 

 

 
 It gets calculated like this:
 

 (A D) - (C B)

 
 We call this combination D
 Which stands for "Determinant"
 No big deal ...
 
 So for ...

 

 
 Now we're going to change the matrix a bit.
 We're going to replace the X coefficients
 with the values the equations are equal to.
 
  3X - 5Y = 6
  2X + Y = 5
 
 This matrix is called Dx.

 

 
 Now combine this one the way we did the other one ...
 

 (6 1) - (5 (-5))

 6 - (-25)

31

 
 Now the good part.
 The value of X that makes both equations true
 is Dx divided by D.
 

 

 
 To find out what Y is equal to,
 go back to the original equation
 and this time replace the Y coefficient in the matrix
 with the equation values.
 Combine this matrix to find Dy.
 
 Then we can find Y.

 

 
 So let's find Y already ...
 

 

 

 
 Let's check it and make sure it worked:
 

 

 
 To review, the equations were:
 
  3X - 5Y = 6
  2X + Y = 5
 
 And the matrices are:
 

 

 
 Now that took a while.
 We would probably not use this method
 to solve problems that are this easy,
 but you should notice something ...
 
 Even though this took a long time to solve,
 the instructions on how to do it were really simple.
 We didn't once have to decide what number to subtract from each side
 or what do divide both sides by,
 or anything like that.
 
 A computer could be programmed
 to solve problems like this using matrix techniques pretty easily.
 To program all of the rules we use in regular algebra
 would take a lifetime.
 
 As the problems get bigger and bigger,
 our regular algebra gets very tough to use.
 The matrix stuff stays about the same
 no matter how big the problems get.
 

   copyright 2005 Bruce Kirkpatrick

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