Question 1 : These four properties can be used to compute determinants of any matrix, using ________ .

Question 2 : Like Lagrange, he made much use of determinants in the ________ .

Question 3 : Under the substitution A ↦ log(A ) , the ________ of A , the above equation yields det(A ) = exp(tr(log(A ))) .

Question 4 : The ________ CRng and Grp are related by two functors GL_{n} and ( )^{×} , where GL_{n} maps any given commutative ring R to the group GL_{n} (R ), and ( )^{×} maps any given commutative ring R to the group R ^{×} .

Question 6 : Here the sum is computed over all ________ σ of the numbers {1, 2, ..., n }.

Question 7 : The determinant of a set of vectors is positive if the vectors form a right-handed ________ , and negative if left-handed.

Question 8 : This identity is used in describing the tangent space of certain matrix ________ .

Question 9 : Therefore det is a ________ between GL_{n} and ( )^{×} .

Question 10 : Alternatively one can perform the ________ if possible or the QR decomposition and find the determinant in a similar fashion.