Determinants
Part 1: Basic Properties
- 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).
- 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?
- Compute det(AB), det(BA), and det(A)det(B).
What do you deduce?
- 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?
- 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.
- 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?
- Enter the matrix K defined in your worksheet.
Check whether K is invertible, and compute det(K).
- 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)?].
- Compute det(A + B) and det(A) + det(B). Repeat
with the matrices R and S. What do you deduce?
- Compute the determinants of A and its transpose.
What do you deduce? Test your conclusion with R and S.
modules at math.duke.edu
