Thread: Physical Intrepretation of cofactors

Is there any physical significance of the matrix terms such as cofactors and minors? I state that this is used for finding the inverse but that is rather an abstract concept and does not motivate the student . Is there any real life applications of these terms.

well, sort of...but it's complicated. suppose you had a 3-dimensional container with straight (and parallel) edges, and you wanted to measure its (oriented) volume.

one natural way to do this, is by picking a "side" to measure the oriented area of, and multiplying by the (oriented) "length" of the 3rd dimension.

this amounts to picking a row (column), that is to say one of our 3 "dimensions" (using the word in both the ordinary sense, and in the sense of a vector space basis),

and expanding by co-factors and minors (the alternating signs in the co-factors are to keep track of orientation, the scalars are to keep track of magnitude).

that is to say, a determinant measures (linear) distortion from a (standard right-hand oriented) unit cube,

and "expansion by minors and co-factors" helps us do it "one dimension at a time".

determinants and inverses are intimiately related, because the determinant is a multiplicative map: det(AB) = det(A)det(B).

this means, in particular, that when A HAS an inverse, det(A)det(A^-1) = det(AA^-1) = det(I) = 1, so det(A^-1) = 1/det(A).

since one can use co-factors and minors to define the determinant, only a slight adjustment is required to use co-factors and minors to define an inverse.

determinants are, in fact, sort of like "linear transformations to the n-th degree", they are an example of a "multi-linear map".

being functions of multiple vectors, rather than just a single one, they are necessarily more involved than matrices (which we can regard as

the "general form" of a linear function of just one variable (vector), the one that takes v to Av, where v is a vector, and A is a matrix).

the computation by hand of a 5x5 determinant is, erm, ugly.

this is a little vague, and imprecise, but is sort of the "gist" of it.

If any physical interpretation were to exist, I would think that it would depend on the physical interpretation of the matrix that you are starting with.