I have a problem that asks:
If A, B, and C are m x n matrices, show that:
A ~ A
If A ~ B then B ~ A
If A ~ B and B ~ C then A ~ C
Now I know the definition of row equivalence is that two matrices are row equivalent if one can be converted into the other using elementary row operations. My question is how do I show this as a proof?
Let's take the easy first one as an example. If I were to say:
Proof:
By the definition of row equivalence, for two matrices to be equivalent, one must be able to be converted into the other using elementary row operations. Therefore it suffices to show row equivalence through elementary row operations.
Using the elementary operation: $R_i -> kR_i$
In particular: A -> 1A = A
Since A can reach itself through elementary operations, A ~ A.
END OF PROOF
Would something like this work? If not what would a formal proof showing row equivalence look like?
Best Answer
According to JMoravitz, the approach to A ~ A is correct.
For the other two it suffices to show that:
Symmetry: show the reverse of each operation, and comment how the reverse sequence of operations are valid and thus means that the statement holds.
Transitivity: Since A ~ B and B ~ C holds then there exists a sequence that goes from A to B and from B to C so if you were to use the sequence from A to B starting at A and adding on the sequence from B to C will yield a sequence of operation that goes from A to C and thus that statement holds.
For a more rigorous proof one may want to take Prasun Biswas' approach explained in the comments.