The (i,j)-minor of a matrix: multilinear map?

[quasar987]

Recall that for an nxn matrix A, the (i,i)-minor of A is defined as $M_{ij}(A)=detA(i|j)$, where A(j|i) stands for the matrix (n-1)x(n-1) obtained from A by removing the ith line and jth column.

Also note that we can view det as a map from R^n x ... x R^n to R taking n vectors from R^n, staking them as the lines of a nxn matrix and taking the determinant of that. And it is a well known fact from linear algebra that this map is n-linear.

In the same way, we can view $M_{ij}$ as a function from R^n x ... x R^n to R, and it is in this context that I ask if the (i,j)-minor of a matrix is a multilinear map.

My book says that it is, but I find it odd that if one multiplies the ith component a_i of $M_{ij}$ by a constant c (for instance 0), one gets not $cM_{ij}(a_1,...,a_i,..,a_n)$ as one should, but rather $M_{ij}(a_1,...,a_i,..,a_n)$ because the ith line is not taken into acount altogether! In other words, $M_{ij}(a_1,...,a_i,..,a_n)$ is independant of a_i !

Am I right?

 

[morphism]

I guess it depends on how exactly you're defining the function M_ij, i.e. does its domain consist of n or n-1 copies of R^n (where in the latter case we're discarding the 'ith component')?

 

[quasar987]

Yes, I suppose this is what the text implied. Ok.