Page 174 - 35Linear Algebra

174 Determinants

Figure 8.2: Remember what row swap does to determinants!

Reading homework: problem 8.2

Applying this result to M = I (the identity matrix) yields

i
det E = −1 ,
j
i
where the matrix E is the identity matrix with rows i and j swapped. It is a row swap
j
elementary matrix.
This implies another nice property of the determinant. If two rows of the matrix
are identical, then swapping the rows changes the sign of the matrix, but leaves the
matrix unchanged. Then we see the following:

Theorem 8.1.2. If M has two identical rows, then det M = 0.

8.2 Elementary Matrices and Determinants

In chapter 2 we found the matrices that perform the row operations involved
in Gaussian elimination; we called them elementary matrices.
0
As a reminder, for any matrix M, and a matrix M equal to M after a
0
row operation, multiplying by an elementary matrix E gave M = EM.

Elementary Matrices

We now examine what the elementary matrices to do determinants.

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