Some properties of the determinant. Exercises 24 and 25 show that it is not difficul...

Some properties of the determinant. Exercises 24 and 25 show that it is not difficult to compute determinants of even large matrices, provided that the matrices have a nice form. The following operations (called elementary row operations) can be used to transform an n × n matrix into one in upper triangular form: I. Exchange rows i and j . II. Multiply row i by a nonzero scalar. III. Add a multiple of row i to row j . (Row i remains unchanged.) For example, one can transform the matrix

into one in upper triangular form in three steps: Step 1. Exchange rows 1 and 2 (this puts a nonzero entry in the upper left corner):

Step 2. Add −1 times row 1 to row 3 (this eliminates the nonzero entries below the entry in the upper left corner):

Step 3. Add row 2 to row 3:

The question is, how do these operations affect the determinant? (a) By means of examples, make a conjecture as to the effect of a row operation of type I on the determinant. (That is, if matrix B results from matrix A by performing a single row operation of type I, how are det A and det B related?) You need not prove your results are correct. (b) Repeat part (a) in the case of a row operation of type III. (c) Prove that if B results from A by multiplying the entries in the ith row of A by the scalar c (a type II operation), then det B = c · det A.