Question

What is the determinant of the matrix 'X' as a function of the determinants of the matrices 'A' and 'B', with A^-1* X = B^-1 ?

Answer 1

what if we multiplied both sides by A,. that would certainly leave only the X on teh LHS

Answer 2

Left-multiplying both sides of the matrix by A, we obtain:\[X=AB^{-1}\]A property of determinants tells us that:\[detX=\det(AB^{-1})=\det(A)\det(B^{-1})\]Recall that A and B must both be invertible matrices for this to hold. Additionally, this implies that the determinant of X will never be zero since the determinants of A and B must be non-zero if they are invertible.