Matrix Operations

Part 4: Determinants

1. For the matrices A and B defined in Part 1, compute det(A) and det(B). Do you notice anything of interest?
2. Calculate the determinants of the identity, zero, and diagonal matrices defined in Part 3. What do you observe?
3. Enter random matrices R and S, and compute det(RS), det(SR), and det(R) det(S). Re-enter these lines a few times. What do you deduce?
4. Recall that the matrix A is invertible. Compute det(A) and det(A^-1). What do you deduce? Repeat with the matrix R to see if your conclusion is the same.
5. Enter a random 4 x 4 matrix P, and define Q to be P^-1AP. Compare det(Q) and det(A). What do you deduce? Why does this follow from your conclusions in steps 3 and 4?
6. Compute det(A + B) and det(A) + det(B). Repeat with the matrices R and S. What do you deduce?
7. Construct a 2 x 2 matrix T with "symbolic" elements T₁₁, T₁₂, etc. -- that is, don't give numeric values to the entries. Evaluate det(T). Repeat for 3 x 3 and 4 x 4. This shows the formulas by which determinants are computed. You may be familiar with the first two of these formulas -- and you may discover why it's a good idea to let the computer evaluate 4 x 4 (and larger) determinants.