Determinants

Part 1: Basic Properties

1. In your worksheet you will find definitions of A and B with which we will carry out some experiments. Enter these definitions, and compute det(A) and det(B).
2. Enter the 4 x 4 and 9 x 9 identity and zero matrices defined in the worksheet. Calculate the determinants of all of these matrices. What do you observe?
3. Compute det(AB), det(BA), and det(A)det(B). What do you deduce?
4. Let's check your conclusion in the preceding step. Enter the random matrices R and S, and compute det(RS), det(SR), and det(R)det(S). Re-execute these commands a few times to get new random matrices and their determinants. Is your conclusion the same?
5. Check that the matrix B is invertible. Compute det(B) and det of the inverse of B. What do you deduce? Repeat with the matrix R to see if your conclusion is the same.
6. Enter the random 4 x 4 matrix P, and define Q to be P^-1 A P. Compare det(Q) and det(A). What do you deduce? Why does this follow from your conclusions in steps 3, 4, and 5?
7. Enter the matrix K defined in your worksheet. Check whether K is invertible, and compute det(K).
8. You now have enough evidence to complete the following statements:
   A square matrix M is invertible if and only if det(M) ... .
   If a matrix M is invertible, then the determinant of the inverse of M is ... [related in what way to det(M)?].
9. Compute det(A + B) and det(A) + det(B). Repeat with the matrices R and S. What do you deduce?
10. Compute the determinants of A and its transpose. What do you deduce? Test your conclusion with R and S.

---

modules at math.duke.edu