



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 D_{x}. 








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
D_{x} 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 D_{y}. 





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 
