Determinants and bilinear forms

If one calculates the product of diagonal elements of the $U$ matrix in a $LUP$ factorization of a given matrix $A$, one can calculate the determinant of $A$. Also it is known that $LUP$ factorization can be reduced to a series of matrix multiplications which are bilinear forms. Does that mean determinant calculation (multilinear form) can be reduced to a bilinear calculation?

• Sorry for my ignorance, but what is bilinear calculation? (I know what a bilinear form is.) Nov 5, 2010 at 2:16
• Hi Tusyoshi I was implying the same. Nov 5, 2010 at 2:35
• Obviously the deterinant is not a bilinear form in the elements of the matrix (no matter how you divide the elements into two sets), if that is what you are asking. Nov 5, 2010 at 3:06
• Hi Tsuyoshi. Yes I know that determinant is a multilinear form. However using only a sequence of matrix multiplications one can reduce to $LUP$. From which taking the product of diagonals of $U$ suffice to find the determinant. Multiplication is a bilinear map. I was curious then may be the determinant is actually a product of bilinear forms. Nov 5, 2010 at 3:24
• I feel this is just a tautological question. Nov 5, 2010 at 3:25