http://www.ck12.org Chapter 8. Systems and Matrices
8.6 Augmented Matrices
Here you will solve systems of equations using augmented matrices.
The reason why the rules for row reducing matrices are the same as the rules for eliminating coefficients when
solving a system of equations is because you are essentially doing the same thing in each case. When you write
and rewrite the equation every time you end up writing down lots of extra information. Matrices take care of this
information by embedding it in the location of each entry. How would you use matrices to write the following
system of equations?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61456http://www.youtube.com/watch?v=BWBckWPjfpw James Sousa: Augmented Matrices: Row Echelon Form
Guidance
In order to represent a system as a matrix equation, first write all the equations in standard form so that the coefficients
of the variables line up in columns. Then copy down just the coefficients in a matrix array. Next copy the variables
in a variable matrix and the constants into a constant matrix.
x+y+z= 9
x+ 2 y+ 3 z= 22
2 x+ 3 y+ 4 z= 31
1 1 1
1 2 3
2 3 4
·
x
y
z
=
9
22
31
The reason why this works is because of the way matrix multiplication is defined.
1 1 1
1 2 3
2 3 4
·
x
y
z
=
1 x+ 1 y+ 1 z
1 x+ 2 y+ 3 z
2 z+ 3 y+ 4 z
=
9
22
31
Notice how putting brackets around the two matrices on the right does very little to hide the fact that this is just a
regular system of 3 equations and 3 variables.