text: Sylvester's theorem equates this function of the identity plus A times B to this function of identity plus
full_question: Sylvester's theorem equates this function of the identity plus A times B to this function of identity plus Btimes A. This function is the unique antisymmetric multilinear form which takes the set of standard basisvectors to one. This quantity can be written as the sum over all permutations sigma of the sign of sigma timesan expansion term, the product of a sub i comma sigma i, which is known as expansion by cofactors. It givesthe area of the parallelogram formed by two vectors. The solutions to a system of linear equations are a ratio oftwo of these quantities, by Cramer's Rule. If this quantity is non-zero, a matrix is invertible. For 10 points,name this quantity that for a two by two matrix is a times d minus b times c.
first_sentence: Sylvester's theorem equates this function of the identity plus A times B to this function of identity plus
raw_answer: determinant