Theorem 3 of Lay Section 32 states how row operations affect

Theorem 3 of Lay Section 3.2 states how row operations affect the determinant of a matrix. This theorem is proven (in the book) using the cofactor definition of the determinant, but they can also be proven (perhaps in a more intuitive way) using the pattern definition of the determinant. For example, in class we discussed why the row operation R_i rightarrow kR_i scales the determinant by k; this is because for every pattern P in the matrix, the result of R_i rightarrow kR_i is to change prod(P) rightarrow kprod(P), while leaving the inversions unchanged. (i) Using the pattern definition, explain why swapping two consecutive rows of a matrix changes the sign of the determinant. (ii) Non-consecutive rows of a matrix can be swapped by performing a series of consecutive row swaps. For example, the operation R_1 doubleheadarrow R_3 is equivalent to the sequence of operations R_1 doubleheadarrow R_2, R_2 doubleheadarrow R_3, R_1 doubleheadarrow R_2. In general, how many row operations does it take to swap row i and row i + k? (iii) Use your answer to the previous question to explain why the row operation R_i doubleheadarrow R_j changes the sign of the determinant.

Solution

(i) Suppose P is a pattern in in a matrix .Now we interchange the two rows of that matrix say ith and jth row

The result of R_i <-> R_j is to change prod(P)-> (-1)prod(Q) will be one inversion

Hence the determinant is multiplied by negative 1

(ii) We will swap R_i <-> R_i+1, R_i+1 <-> R_i+2, ....R_i+j <-> R_i+k, R_i+k<->R_i

So it takes (k+1) row operations to swap R_i to R_i+k