Left, Right Inverses

  • Thread starter psholtz
  • Start date
  • #1
136
0

Main Question or Discussion Point

Suppose we have a linear transformation/matrix A, which has multiple left inverses B1, B2, etc., such that, e,g,:

[tex]B_1 \cdot A = I[/tex]

Can we conclude from this (i.e., from the fact that A has multiple left inverses) that A has no right inverse?

If so, why is this?
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,793
922
Suppose there were a right inverse, say, R. Multiplying the equation [itex]B_1A= I[/itex] on the right by R gives [itex](B_1A)R= IR[/itex] so that [itex]B_1(AR)= B_1= R[/itex] But then doing the same with [itex]B_2A[/itex] leads to [itex]B_2= R[/itex].

In other words, if a matrix has both right and left inverses, then it is invertible and both right and left inverses are equal to its (unique) inverse.
 

Related Threads for: Left, Right Inverses

Replies
3
Views
3K
  • Last Post
Replies
3
Views
8K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
3
Views
4K
Replies
2
Views
636
  • Last Post
Replies
2
Views
2K
Replies
9
Views
2K
Top