Now we start to use the determinant. Understanding the cofactor formula allows us to show that A-1 = (1/detA)CT, where C is the matrix of cofactors of A. Combining this formula with the equation x = A-1b gives us Cramer's rule for solving Ax = b. Also, the absolute value of the determinant gives the volume of a box.
Cramer's Rule, Inverse Matrix and Volume
Determinants and Volume
Work the problems on your own and check your answers when you're done.