Finding the Inverse Matrix for a Finite Set Relation R

[brad sue]

Hi .
I have this question( discrete math) :
How can the matrix for R^-1 , the inverse of the relation R, be found from the matrix representing R, when R is a relation a finite set A.

How can I do this problem?

 

[HallsofIvy]

When in doubt, try a simple example. Suppose A= {1, 2, 3} and R is defined as {(1, 1), (1, 3), (2, 3)} (I just made that up pretty much at random. Remember that a "relation on A" is just a collection of pairs of members of A.) Now, the "matrix representing R" is the matrix having 1 in the "a row, b column" when (a,b) is in R, 0 otherwise. here, labeling the rows and columns 1, 2, 3 in that order, the matrix is
\left(\begin{array}{ccc}1 & 0 & 1\\0 & 0 &1 \\0 & 0 & 0\end{array}\right).

What is the relation R^-1? What matrix represents it? How are the two matrices related?

 

[brad sue]

What is the relation R^-1? What matrix represents it? How are the two matrices related?

I think, we need to find a matrix R^-1 such that R*R^-1=indentity matrix