Page 179 - 35Linear Algebra

8.2 Elementary Matrices and Determinants 179

Figure 8.4: Adding one row to another leaves the determinant unchanged.

We also have learnt that

i

det S (µ)M = det M .
j
Notice that if M is the identity matrix, then we have

i
i
det S (µ) = det(S (µ)I) = det I = 1 .
j j

8.2.4 Determinant of Products

In summary, the elementary matrices for each of the row operations obey

i
E i j = I with rows i, j swapped; det E = −1
j
i
i
R (λ) = I with λ in position i, i; det R (λ) = λ
i
i
S (µ) = I with µ in position i, j; det S (µ) = 1
j
j
Elementary Determinants

Moreover we found a useful formula for determinants of products:

i
i
i
Theorem 8.2.1. If E is any of the elementary matrices E , R (λ), S (µ),
j j
then det(EM) = det E det M.
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