Interesting Trick To Remember
Point of Post: Often problems like:
” Determine whether the matrix
This post gives a trick, which when applicable, can make monstrosities like the above into simple calculations.
So, as the above motivated we are looking for easy to check conditions which will tell us when a matrix, like the above, is invertible. We could check the rank, the span of the columns etc., the spectrum (multiset of eigenvalues), etc. But, in a time-oriented setting (e.g. a math competition) these conditions aren’t so feasible. But, even given all the time in the world checking the column space of a matrix is not often easy, and it would be preferable to consider another means. So, we fall back on our old friend, the determinant. The quantitative measure of whether a matrix is invertible. But, this in itself may be a herculean task if the numbers are as “messy” as those in the above matrix. But, when we are considering matrices with integer entries, there is a neat “trick” one can check that can often (not always) make our lives easier. It stems from two observations: first
Observation: An integer is zero if and only if for every
Put differently, if and only if for all . And the second
Observation: If is defined to be the remainder of upon division by ,
The second of these observations is clear, but the first may require some explaining. What this really means is that every integer can be uniquely represented as where and (this is really just the division algorithm) and so we’re saying that . So, with this definition we can prove the above by noticing that if that
from where it becomes clear. But, with this definition of we may restate the first observation as ” if and only if for all “.
So now, with these observations made we may begin our hunt for easier (in a sense) conditions for when an integer entried matrix is invertible. We begin by recalling that if (where is the ring of matrices with entries in )
that is defined by
And a necessary and sufficient condition for to be invertible is that . But, note that since our observations tell us that is not invertible if and only if for all . But,
So that if we define
then what we’ve really shown is that is not invertible if and only if for all . At this point you’re probably asking “And this is easier than just taking the determinant…how?”
Well, let me show you how. Consider the matrix we had in the motivation, call it . Then,
and with this it is much easier to find that
But, this means that must be invertible!
Thus, the use of this is mostly when one wants to prove that something is invertible. If one can find any for which if one reduces the entires of a matrix , takes the determinant, and then reduces the result that one doesn’t get then the matrix must be invertible. Probably the most applicable (as in the above example) case is the following corollary
Corollary: If a matrix has the quality that is odd, then is invertible.
References: I haven’t found a book that explicitly states this theorem, but for more information on the determinant see
1. Horn, Roger A., and Charles R. Johnson. Matrix Analysis. Cambridge [u.a.: Cambridge Univ., 2009. Print.
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