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8.2 Elementary Matrices and Determinants 177
Figure 8.3: Rescaling a row rescales the determinant.
Thus, multiplying a row by λ multiplies the determinant by λ. I.e.,
det R i (λ)M = λ det M .
Since R i (λ) is just the identity matrix with a single row multiplied by λ,
then by the above rule, the determinant of R i (λ) is λ. Thus
⎛
1
det R i (λ) = det
⎜
⎝
. ..
λ
⎞
= λ ,
. ⎟ .. ⎠
1
and once again we have a product of determinants formula
det ( R i (λ)M ) = det ( R i (λ) ) det M.
8.2.3 Row Addition
The final row operation is adding µR j to R i . This is done with the elementary
matrix S i j(µ), which is an identity matrix but with an additional µ in the i, j
position;
177