Page 55 - 35Linear Algebra

2.3 Elementary Row Operations 55

Example 25 (Collecting EROs that undo a matrix)

   
0 1 1 1 0 0 2 0 0 0 1 0
2 0 0 0 1 0 ∼ 0 1 1 1 0 0
   
0 0 1 0 0 1 0 0 1 0 0 1

 1   1 
1 0 0 0 0 1 0 0 0 0
2 2
∼  0 1 1 1 0 0  ∼  0 1 0 1 0 −1  .
0 0 1 0 0 1 0 0 1 0 0 1
As we changed the left side from the matrix M to the identity matrix, the
right side changed from the identity matrix to the matrix which undoes M.

Example 26 (Checking that one matrix undoes another)

 1     
0 0 0 1 1 1 0 0
2
1 0 −1 2 0 0 = 0 1 0 .
     
0 0 1 0 0 1 0 0 1
If the matrices are composed in the opposite order, the result is the same.

   1   
0 1 1 0 0 1 0 0
2
2 0 0 1 0 −1 = 0 1 0 .
     
0 0 1 0 0 1 0 0 1
Whenever the product of two matrices MN = I, we say that N is the
inverse of M or N = M −1 . Conversely M is the inverse of N; M = N −1 .

In abstract generality, let M be some matrix and, as always, let I stand
for the identity matrix. Imagine the process of performing elementary row
operations to bring M to the identity matrix:

(M|I) ∼ (E 1 M|E 1 ) ∼ (E 2 E 1 M|E 2 E 1 ) ∼ · · · ∼ (I| · · · E 2 E 1 ) .

The ellipses “· · · ” stand for additional EROs. The result is a product of
matrices that form a matrix which undoes M

· · · E 2 E 1 M = I .

This is only true if the RREF of M is the identity matrix.

Definition: A matrix M is invertible if its RREF is an identity matrix.

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