Determinants
Part 2: Row and Column Operations
- Interchange any two rows of the matrix A, and compute the determinant of the resulting matrix. What do you observe?
Try this experiment with two columns of A.
- Let E be the matrix obtained from A by multiplying
one row of A by 2. Compute det(E), and compare with det(A).
What property of determinants have you demonstrated? Try this experiment
with a column of A.
- Let F = 2*A. Compute det(F) and det(F)/det(A).
What do you deduce? Reconcile this result with your conclusion in the preceding
step.
- Display A again. Add a non-zero multiple of
one row of A to another row, and compute the determinant of the
resulting matrix. What property of determinants have you demonstrated?
Also try this experiment with adding a non-zero multiple of one column
to another column.
- Enter the matrices U and L defined in the worksheet,
and compute the determinant of each. How could you have done these computations
in your head?
- Enter the random 4 x 4 matrix T. Now replace
one row of T with all zeros (multiply one row by zero). Compute the determinant
of this new T. Next, replace the row of zeros with a non-zero multiple
of another row. Compute det(T) again. What do you conclude?
- Repeat the experiments in the preceding step
with a column instead of a row.
modules at math.duke.edu
