Page 41 - 35Linear Algebra

2.1 Gaussian Elimination 41

called the Identity Matrix, since this would give the simple statement of a
solution x = a, y = b. The same goes for larger systems of equations for
which the identity matrix I has 1’s along its diagonal and all off-diagonal
entries vanish:

 
1 0 · · · 0
0 1 0
 
I =  . . . . . . 


 . . . 
0 0 · · · 1

Reading homework: problem 2

For many systems, it is not possible to reach the identity in the augmented
matrix via Gaussian elimination. In any case, a certain version of the matrix
that has the maximum number of components eliminated is said to be the
Row Reduced Echelon Form (RREF).

Example 13 (Redundant equations)

) ! ! (
x + y = 2 1 1 2 1 1 2 x + y = 2
⇔ ∼ ⇔
2x + 2y = 4 2 2 4 0 0 0 0 + 0 = 0

This example demonstrates if one equation is a multiple of the other the identity
matrix can not be a reached. This is because the first step in elimination will make
the second row a row of zeros. Notice that solutions still exists (1, 1) is a solution.
The last augmented matrix here is in RREF; no more than two components can be
eliminated.

Example 14 (Inconsistent equations)

) ! ! (
x + y = 2 1 1 2 1 1 2 x + y = 2
⇔ ∼ ⇔
2x + 2y = 5 2 2 5 0 0 1 0 + 0 = 1

This system of equation has a solution if there exists two numbers x, and y such that
0 + 0 = 1. That is a tricky way of saying there are no solutions. The last form of the
augmented matrix here is the RREF.

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