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.
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