Cramer’s rule grows like n! with the size n of the matrix. In this problem, you will construct the key steps of the proof of this scaling. The limiting step for Cramer’s rule is to compute the determinants of the matrices. So you want to show that the effort to compute the determinant of a matrix by cofactor expansion is factorial in the number of rows (columns) n of the matrix. To arrive at this answer, you should make the following explicit calculations.
(a) Determine the number of operations (addition, subtraction, multiplication, division) required to compute the determinant of a 2×2 matrix. Call this number q for future reference.
(b) Repeat for a 3 × 3 matrix using co-factor expansion. You can ignore the effort required to set the sign of the coefficients. Report the answer in terms of q.
(c) Repeat for a 4 × 4 matrix using co-factor expansion. You can ignore the effort required to set the sign of the coefficients. Report the answer in terms of the size of the matrix n = 4 and q.
(d) Repeat for a 5 × 5 matrix using co-factor expansion. Report the answer in terms of the size of the matrix n = 5 and q.
(e) Use the formula from the last step to estimate the number of operations required for a large n × n matrix using co-factor expansion. You only need to report the term that dominates as n → ∞.
