Page 368 - 35Linear Algebra

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equivalent to the system of equations

x + y = 27
2x − y = 0 ?

Well the augmented matrix is just a new notation for the matrix equation

1 1 x 27
=
2 −1 y 0
and if you review your matrix multiplication remember that

1 1 x x + y
=
2 −1 y 2x − y
This means that

x + y 27
= ,
2x − y 0
which is our original equation.

Equivalence of Augmented Matrices

Lets think about what it means for the two augmented matrices

1 1 27 1 0 9
and
2 −1 0 0 1 18
to be equivalent: They are certainly not equal, because they don’t match in
each component, but since these augmented matrices represent a system, we
might want to introduce a new kind of equivalence relation.
Well we could look at the system of linear equations this represents

x + y = 27
2x − y = 0

and notice that the solution is x = 9 and y = 18. The other augmented matrix
represents the system
x + 0 · y = 9
0 · x + y = 18

This clearly has the same solution. The first and second system are related
in the sense that their solutions are the same. Notice that it is really
nice to have the augmented matrix in the second form, because the matrix
multiplication can be done in your head.

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