Finding the Inverse Matrix for a Finite Set Relation R

In summary, The process of finding the matrix for R-1, the inverse of a relation R, from the matrix representing R, involves finding a matrix R-1 such that R*R-1 is equal to the identity matrix of the same order. This can be done by using a simple example and labeling the rows and columns accordingly. The two matrices are related in that they are multiplicative inverses of each other.
  • #1
brad sue
281
0
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?
 
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  • #2
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
[tex]\left(\begin{array}{ccc}1 & 0 & 1\\0 & 0 &1 \\0 & 0 & 0\end{array}\right)[/tex].

What is the relation R-1? What matrix represents it? How are the two matrices related?
 
  • #3
HallsofIvy said:
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
 

What is an inverse matrix in a relation?

An inverse matrix in a relation is a matrix that when multiplied by the original matrix, results in an identity matrix (a square matrix with 1s on the main diagonal and 0s everywhere else). In other words, it "undoes" the effects of the original matrix.

Why is the inverse matrix important in relation to matrices?

The inverse matrix is important because it allows us to solve equations involving matrices, which is useful in many fields such as engineering, physics, and economics. It also helps us find solutions to systems of linear equations and perform other mathematical operations with matrices.

How is the inverse matrix calculated?

The inverse matrix is calculated by using the Gauss-Jordan elimination method or the adjugate matrix method. Both methods involve a series of mathematical operations on the original matrix to transform it into the identity matrix.

Under what conditions does a matrix have an inverse?

A matrix has an inverse if it is a square matrix (with the same number of rows and columns) and its determinant (a special value calculated from the elements of the matrix) is not equal to zero. If the determinant is zero, the matrix is said to be singular and does not have an inverse.

Can a matrix have more than one inverse?

No, a matrix can only have one inverse. This is because the inverse of a matrix is unique and can be calculated using the same method regardless of the specific values in the matrix. If a matrix appears to have more than one inverse, it is likely due to a calculation error.

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